0Why this page, and how to use it

Analysis builds numbers up from a handful of rules. To follow that, you need to know what a "rule for numbers" is — that is what fields and ordered fields are — and how mathematicians argue with such rules — that is what proofs and quantifiers are. If you already know these topics from Diskrete Strukturen or Lineare Algebra, skim the summary at the end and do the self-test. If not, read the page top to bottom once; it takes about an hour.

B.1Sets and notation [Mengen]

Intuition. A set is a collection of objects, called its elements. Order and repetition do not matter: \(\{1,2,3\}=\{3,1,2\}=\{1,1,2,3\}\). A set is completely determined by which elements it contains.

Reading sums and products: \(\sum\) and \(\prod\) Extra

B.2Logic and proof techniques [Beweistechniken]

Four proof techniques cover almost every proof in the course:

Quantifiers (\(\forall\), \(\exists\)) and how to negate them are explained at the start of Chapter 1 (§1.1 · Quantifiers); induction is explained with Bernoulli's inequality in Chapter 2. "o.B.d.A." [ohne Beschränkung der Allgemeinheit] = "without loss of generality": the other case works the same way.

B.3Maps [Abbildungen, Funktionen]

Intuition. A map \(f:A\to B\) is a rule that assigns to every element \(a\) of \(A\) exactly one element \(f(a)\) of \(B\). Picture arrows: every element of \(A\) sends out exactly one arrow into \(B\).

B.4Groups, rings and fields [Gruppen, Ringe, Körper]

Why this matters. When you simplify \(\frac{2x+6}{2}=x+3\) or move a term to the other side of an equation, you use rules like "you may divide by any nonzero number". A field is exactly a set with \(+\) and \(\cdot\) in which all the familiar rules of school algebra hold. Saying "ℝ is a field" is the script's way of saying "all of that works in ℝ". Rings and groups are the same idea with fewer rules — useful to see which rule fails where.

All three are built from binary operations [Verknüpfungen]: rules that combine two elements of a set \(M\) into a third element of \(M\), like \(+\) on \(\mathbb Z\).

Group [Gruppe]

Ring [Ring]

Field [Körper]

What follows from the field axioms

Everything you do in algebra follows from the axioms. Here are the rules the course uses silently, each with its two-line proof — they show how "obvious" rules are actually consequences of the axioms.

B.5Ordered fields [angeordnete Körper]

Intuition. A field says how to calculate. An ordering adds the notion of "positive", and with it \(<\). Analysis lives on inequalities (\(|a_n-a|<\varepsilon\)), so this is the second half of what makes ℝ suitable.

B.6Powers and roots [Potenzen und Wurzeln]

Intuition. \(a^n\) starts as a shorthand: "multiply \(a\) by itself \(n\) times". Everything else — \(a^0\), \(a^{-3}\), \(a^{1/2}\), \(a^{\sqrt2}\) — is a later extension, and each extension was chosen for one reason: so that the five rules in the second box below keep holding. That is worth knowing, because it explains every restriction you are about to see. The rules are easy; what costs points is the domains — which rule needs \(a\ge0\), which needs \(a>0\), and why \(\sqrt{x^2}\) is not \(x\).

The course makes all of this rigorous much later: the \(n\)-th root exists by Anwendung 4.13 (intermediate value theorem), and the general power \(x^a:=e^{a\ln x}\) is Definition 5.11. This section is the algebra you are assumed to have from school, stated with the domains the course silently relies on.

B.7Reading the exam in German [Prüfungssprache]

This guide is in English, but the exam paper is in German. Most of the mathematical vocabulary appears in brackets throughout the chapters; what is easy to overlook are the instruction words and the sentences that tell you how much you have to write. Those decide your points as much as the mathematics does.

ΣSummary

1. Sets: read \(\in,\subseteq,\cup,\cap,\setminus,\times\) and \(\{x\in A:\dots\}\); prove \(A=B\) by showing both inclusions.
2. Proofs: \(A\Rightarrow B\) is not \(B\Rightarrow A\); the contrapositive \(\neg B\Rightarrow\neg A\) is the same statement; contradiction and induction are the other two standard tools.
3. Maps: injective = no collisions, surjective = everything hit, bijective = both = invertible. Domain and codomain are part of the map.
4. Structures: group (one operation, invertible) → ring (+, ·, no division) → field (division by nonzero), each adding rules to the previous one. ℤ is a ring, ℚ, ℝ, ℂ are fields.
5. Consequences of the field axioms: \(a\cdot0=0\), \((-1)(-1)=1\), \(ab=0\Rightarrow a=0\) or \(b=0\).
6. Order: "positive" with rules (a), (b) ⇒ all inequality rules; multiplying by a negative flips. ℚ, ℝ ordered; ℂ, \(\mathbb Z/p\) not. ℝ = the complete ordered field.
7. Sums: \(\sum\) is linear, splits, shifts (\(j=k-1\)) and telescopes; empty sum \(=0\), empty product \(=1\). A product of two sums is not the sum of the products.
8. Powers and roots: the five rules (\(a^xa^y=a^{x+y}\), \(\frac{a^x}{a^y}=a^{x-y}\), \((a^x)^y=a^{xy}\), \((ab)^x=a^xb^x\), \(\left(\frac ab\right)^x=\frac{a^x}{b^x}\)) hold for \(a,b>0\) and all real exponents, and for \(a,b\neq0\) with integer exponents. They never apply to a sum: \(\sqrt{a+b}\neq\sqrt a+\sqrt b\). \(\sqrt{x^2}=|x|\); fractional powers need a positive base; \(f^g=e^{g\ln f}\) is the way past a variable exponent.
9. Exam German: zeigen/beweisen = full proof, begründen = give the reason, angeben = answer only; watch for "nur das Ergebnis wird bewertet" and "alle Zwischenschritte sind zu begründen".

?Self-test

0The big picture

Every new number set in this chapter exists because the previous one had a problem. Reading the chain left to right is the storyline of Section 1.1:

The step that matters most for analysis is ℚ → ℝ. Analysis is about limits, and you can only take limits if they actually land somewhere. In ℚ, the numbers 1, 1.4, 1.41, 1.414, … creep towards a point that is not there, namely √2. The real numbers fill all such holes. The precise statement of "no holes" is the supremum axiom in 1.2, and it is the foundation for everything in Chapters 2–4.

So the chapter has three parts:

1. 1.1 Which number sets exist, how they differ (countable vs. uncountable, ordered vs. not ordered).
2. 1.2 What makes ℝ special: sup/inf exist, no infinitely large or small numbers, fractions are everywhere.
3. 1.3 The inequalities you will use to estimate things in every later proof: triangle inequality and Cauchy–Schwarz.

1.1Number sets at a glance [Zahlenmengen im Überblick]

Quantifiers [Quantoren]

Before any numbers, the script introduces two symbols you will read thousands of times:

• \(\forall x\): "for all \(x\)" [für alle]
• \(\exists x\): "there exists an \(x\)" [es existiert]

The single most useful skill with them is negation, because every proof by contradiction starts by negating a statement. The rule: flip every \(\forall\) into \(\exists\), every \(\exists\) into \(\forall\), and negate the statement at the end.

The order of quantifiers matters. "\(\forall\) people \(\exists\) a mother" is true; "\(\exists\) a mother \(\forall\) people" (one mother of everyone) is false. You will see this again in the definition of convergence, where \(n_0\) is allowed to depend on \(\varepsilon\).

The number sets

"Field" just means: you can add, subtract, multiply and divide (except by 0), and all the usual rules hold (commutative, associative, distributive, 0 and 1 behave as expected). ℚ is also a field; ℤ is not, since \(1/2\notin\mathbb Z\). The precise axioms, what follows from them (like \((-1)(-1)=1\) and "\(ab=0\Rightarrow a=0\) or \(b=0\)"), and how rings and groups differ are on the Basics page.

√2 is not rational

This is the proof that ℚ has holes, so it is the reason ℝ is needed at all.

Countable and uncountable [abzählbar, überabzählbar]

Intuition. A set is countable if you can write its elements in one infinite list \[f(1),\ f(2),\ f(3),\ \dots\] so that every element of the set turns up somewhere in the list — maybe late, but at a definite, finite position. "Countable" means "can be numbered 1st, 2nd, 3rd, …, and nobody is left out".

Part 1 — Why ℚ is countable: Cantor's diagonal walk Extra (the script's "Übung")

The problem. Between any two fractions there are infinitely many others, so you cannot list fractions "in order of size" — there is no next fraction after 0. A naive list like \(\frac11,\frac12,\frac13,\frac14,\dots\) never ends, so it never reaches \(\frac21\). We need a cleverer order.

The idea: put the fractions in a grid. Every positive fraction is \(\frac pq\) with \(p,q\in\mathbb N\) (numerator \(p\), denominator \(q\)). Write \(\frac pq\) in column \(p\), row \(q\):

Every positive rational number appears in this grid (usually several times: \(\frac12=\frac24=\frac36\)). Walking along a row or a column fails, since each is infinite. Instead, walk along the short diagonals:

1. Group the grid by diagonals. Diagonal number \(d\) contains all fractions with \(p+q=d\). Diagonal 2 is just \(\frac11\); diagonal 3 is \(\frac12,\frac21\); diagonal 4 is \(\frac13,\frac22,\frac31\); in general diagonal \(d\) has exactly \(d-1\) entries. Each diagonal is finite — that is the whole point.
2. List the diagonals one after the other, each from its left end to its right end: \[\underbrace{\tfrac11}_{d=2},\ \underbrace{\tfrac12,\ \tfrac21}_{d=3},\ \underbrace{\tfrac13,\ \tfrac22,\ \tfrac31}_{d=4},\ \underbrace{\tfrac14,\ \tfrac23,\ \tfrac32,\ \tfrac41}_{d=5},\ \dots\] Because every diagonal is finite, we finish it and move on to the next — we never get stuck.
3. Every fraction is reached after finitely many steps. Take any \(\frac pq\). It lies on diagonal \(d=p+q\). Before that diagonal come the diagonals \(2,3,\dots,d-1\), with \(1+2+\dots+(d-2)=\frac{(d-2)(d-1)}2\) entries in total, and inside its own diagonal \(\frac pq\) is entry number \(p\). So it appears at position \[n(p,q)=\frac{(p+q-2)(p+q-1)}2+p,\] a definite natural number. Example: \(\frac23\): \(d=5\), so \(n=\frac{3\cdot4}2+2=8\). Check against the list above: \(\frac11,\frac12,\frac21,\frac13,\frac22,\frac31,\frac14,\frac23\) — eighth. ✓
4. Add 0 and the negatives with the ℤ trick: put 0 first and each negative right after its positive twin: \[0,\ \tfrac11,\ -\tfrac11,\ \tfrac12,\ -\tfrac12,\ \tfrac21,\ -\tfrac21,\ \tfrac13,\ -\tfrac13,\ \dots\] If \(r>0\) is at position \(n\) in the positive list, then \(r\) is at position \(2n\) and \(-r\) at position \(2n+1\) here. Every rational number is hit, so this is a surjection \(f:\mathbb N\to\mathbb Q\). ∎

What about the repeats? \(\frac22\) is the same number as \(\frac11\), so 1 appears more than once. The script's definition only asks for a surjective map, so repeats are completely fine. (If you want each number exactly once — a bijection — just skip every fraction that is not fully cancelled, like \(\frac22\) or \(\frac24\). The widget can do both.)

Part 2 — Why ℝ is not countable: Cantor's diagonal argument

Strategy (proof by contradiction): assume someone hands us a list that supposedly contains every number in \([0,1]\). We then construct, from the list itself, a number in \([0,1]\) that is guaranteed not to be in it. So no list can be complete.

A computer-science application from the script:

Ordering [Anordnung]

ℝ is not just a field, it is an ordered field. The script defines "ordered" through a single predicate "\(a>0\)" (being positive) with two rules:

From these two rules alone you can prove that every nonzero square is positive: if \(a>0\) then \(a\cdot a>0\) by (b); if \(-a>0\) then \(a^2=(-a)(-a)>0\), again by (b). That little fact is the key to the next result.

Complex numbers [komplexe Zahlen]

Intuition: a complex number is a point in the plane. We just add a rule for multiplying points.

You don't need to memorise the multiplication formula. Just expand the brackets as usual and replace \(i^2\) by \(-1\):

Exercise Sheet 1 adds three tools you will use constantly, especially in Chapter 5 (Euler's formula):

1.2Properties of the real numbers [Eigenschaften der reellen Zahlen]

Upper bounds, supremum and infimum

Intuition: an upper bound is a ceiling for a set, a number that nothing in the set exceeds. A set has many ceilings. The supremum is the lowest ceiling, the one that touches the set.

The script lists some conventions for the cases the axiom doesn't cover: \(\sup M=\infty\) if \(M\) is not bounded above, \(\inf M=-\infty\) if \(M\) is not bounded below, \(\sup\emptyset=-\infty\), and \(-\infty<a<\infty\) for every real \(a\).

Why ℚ is not enough

This is the precise meaning of "ℝ has no holes": whenever a set pushes upward against a ceiling, the lowest ceiling actually exists as a real number. In ℚ, the lowest ceiling of \(M\) would be √2, and that spot is empty.

How to prove that a number is the supremum

This is what exams ask for, so here is a reusable recipe. It follows directly from the definition ("least upper bound" = an upper bound, and nothing smaller is one). The script uses exactly this reasoning inside its proofs of Satz 1.9 and Satz 2.17.

The Archimedean property [archimedisch]

Decimal representation [Dezimaldarstellung]

Every real number has a decimal expansion \(\pm d_0.d_1d_2d_3\ldots\) with \(d_0\in\mathbb N_0\) and digits \(d_i\in\{0,\dots,9\}\). The script connects decimals to the supremum in both directions:

• Decimal → real number: \(d_0.d_1d_2\ldots\) means \(\sup\{d_0.d_1\ldots d_n: n\in\mathbb N\}\). Every finite truncation is rational (\(1.37=\frac{137}{100}\)), and the real number is the supremum of these truncations.
• Real number → decimal: for \(a\ge0\) set \(p_n=\max\{p\in\mathbb N_0: p\le10^na\}\), and \(\frac{p_n}{10^n}\) gives the first \(n\) digits.

What you must know for 1.2 (script: "Minimale Inhalte"): supremum, infimum, minimum and maximum: know them and be able to determine them.

1.3Important inequalities [Wichtige Ungleichungen]

Why inequalities? From Chapter 2 on, almost every proof reads "make \(|a_n-a|\) smaller than \(\varepsilon\)". You rarely compute \(|a_n-a|\) exactly. You estimate it, and these inequalities are the estimating tools.

Triangle inequality [Dreiecksungleichung]

Young's ε-inequality

Vectors in ℝⁿ

Cauchy–Schwarz inequality

Norms: different ways to measure length

What you must know for 1.3 (script: "Minimale Inhalte"): be able to apply the different variants of the triangle inequality and the Cauchy–Schwarz inequality.

!What the exam asks from Chapter 1

I went through all nine uploaded exams (WiSe18 – WiSe24). Chapter 1 appears in five of them, and almost always in the same form. Here is the complete list:

Conclusion: the one skill you must have cold is determining and justifying sup, inf, max and min. Countability, Cantor, Archimedes and the inequalities are not asked as standalone tasks — but Archimedes and the triangle inequality are used inside nearly every ε-proof of Chapter 2, so you need them anyway. The four tasks not yet worked above follow (the other exam sets are worked in the explorer).

Drill: sup, inf, max, min

Ten sets of the kind the exam uses. Decide the four values before you click — and, for each bound, which of the two steps of the recipe you would write down.

ΣSummary sheet

The whole chapter as one chain of reasoning:

1. ℚ is a perfectly good field, but it has holes: √2 ∉ ℚ (Lemma 1.1), and \(\{a\in\mathbb Q:a^2\le2\}\) has no supremum in ℚ (Remark 1.8).
2. ℝ closes these holes. That is the supremum axiom: nonempty + bounded above ⇒ sup exists.
3. Filling the holes makes ℝ much bigger: ℚ is countable, ℝ is uncountable (Cantor, Lemma 1.3). So almost no real number is computable (Lemma 1.11).
4. From the supremum axiom: ℝ is Archimedean (Satz 1.9) ⇒ for every \(a>0\) some \(\frac1n<a\) ⇒ ℚ is dense (Satz 1.10) ⇒ every real is approximated by fractions / decimals.
5. ℝ is ordered; ℂ = ℝ² gains \(i^2=-1\) but loses the order (Lemma 1.4). In ℂ only \(|z|\) can be compared.
6. To estimate distances, use the triangle inequality (1.12), its reverse (1.13), Cauchy–Schwarz (1.15) and the triangle inequality in ℝⁿ (1.14). These are the tools for all ε-proofs from Chapter 2 on.

Exam focus

Sup/inf/max/min of a set is the Chapter 1 exam task (five of nine exams, see "What the exam asks" above). Every exam set is in the explorer and fully worked on this page.

?Self-test

Click an answer. You get an explanation either way.

0The big picture

Chapter 1 gave us ℝ without holes. Chapter 2 is where that pays off. The storyline:

1. Definition of convergence with ε and \(n_0\): "eventually the terms stay as close to \(a\) as you like". Everything else rests on this.
2. Limit rules (sum, product, quotient) and the squeeze theorem: with these you almost never need ε again when computing limits.
3. Improper convergence (to ±∞) and asymptotic equality (\(a_n\simeq b_n\)): the language for "how fast things grow".
4. Monotone sequences: a monotone sequence converges exactly when it is bounded, and its limit is its sup (or inf). This is the supremum axiom of Chapter 1 at work. It gives us \(e\) and shows that the harmonic numbers \(H_n\) grow to ∞.

2.1Convergence [Konvergenz]

Sequences often come from recursions [Rekursionen], where each term is computed from the previous one, the way a loop computes its next value. Example (Skript 2.2): \(a_0=1,\ a_{n+1}=q\,a_n\) gives the explicit formula \(a_n=q^n\). What happens for large \(n\) depends on \(q\): for \(q=2\) the terms grow to ∞, for \(q=\frac12\) they shrink to 0, and for \(q=-1\) they jump between \(-1\) and \(1\) forever.

The definition of convergence

Intuition. "\(a_n\to a\)" means: pick any tolerance ε, however tiny. From some index on, all terms lie within ε of \(a\). A finite number of early terms may do anything; only the "tail" of the sequence matters.

Read the quantifiers as a game. Your opponent picks ε. You answer with an \(n_0\), which may depend on ε. You win if every term from \(n_0\) on is within ε of \(a\). The sequence converges if you can win against every ε.

Rules for limits [Rechenregeln für Grenzwerte]

Improper convergence [uneigentliche Konvergenz]

Same game as before, but now the opponent names a height \(K\) and you must show that the tail stays above it. A real sequence therefore has three possibilities: it converges to some \(a\in\mathbb R\), it diverges to \(\pm\infty\), or neither (like \((-1)^n\), or \((-1)^nn^2\)).

Asymptotic equality [asymptotisch gleich]

Intuition: the relative error goes to 0. So \(n+1\simeq n\), \(n^2+7n+13\simeq n^2\), and any polynomial is \(\simeq\) its leading term (Skript 2.11). Only the large terms matter; finitely many terms play no role.

Boundedness, inequalities, squeeze theorem

What you must know for 2.1 (script: "Minimale Inhalte"): the definition of convergence and improper convergence with quantifiers; asymptotic equality; computing limits; the squeeze theorem.

2.2Monotone sequences [Monotone Folgen]

How to check monotonicity in practice: look at the sign of \(a_{n+1}-a_n\), or (for positive terms) compare \(\frac{a_{n+1}}{a_n}\) with 1.

Why this theorem is so important: it proves a limit exists without knowing its value in advance. Most limits in this course (\(e\), series with positive terms, Heron's method) are found this way. It is also the reason ℚ would not work: in ℚ, the increasing bounded sequence 1, 1.4, 1.41, … has no limit.

Three tools: Bernoulli, binomial theorem, geometric sum

Euler's number \(e\)

Harmonic numbers and Quicksort

What you must know for 2.2: Satz 2.17 (monotone convergence), the binomial theorem and the geometric sum formula, and the definition of \(e\).

Mixed exercises from the script [Kapitel 13: Verschiedene Aufgaben]

Chapter 13 of the script is a pool of exam-style exercises with full solutions. Two of them are limits of the kind this chapter is about; they are worth doing now, because exams love exactly this format.

!What the exam asks from Chapter 2

Across the nine exams, Chapter 2 appears in five (SoSe19, WiSe19, WiSe22, SoSe23, WiSe23), always as one of five task types. Every one is worked on this page:

Drill: which tool, and what is the limit?

Ten items in exam format. Decide the answer and the rule you would name in the exam.

ΣSummary sheet

1. \(a_n\to a\iff\forall\varepsilon>0\ \exists n_0\ \forall n\ge n_0:\ |a_n-a|<\varepsilon\). Limits are unique; constants in front of ε don't matter; only the tail counts.
2. Compute limits with the rules (only when both limits exist and are finite, denominator ≠ 0), by dividing by the highest power, with the conjugate trick for √-differences, and with the squeeze theorem.
3. \(a_n\to\infty\iff\forall K\ \exists n_0\ \forall n\ge n_0:\ a_n\ge K\). To show it, bound \(a_n\) from below by something that goes to ∞.
4. \(a_n\simeq b_n\iff a_n/b_n\to1\): replace sums by their largest term. It says nothing about \(a_n-b_n\).
5. Convergent ⇒ bounded. Null sequence × bounded ⇒ null sequence.
6. Monotone + bounded ⇒ convergent, limit = sup (increasing) or inf (decreasing). For recursions: first existence, then pass to the limit in the recursion.
7. \(e=\lim(1+\frac1n)^n\in[2.7,3]\); \(H_n\to\infty\) (slowly, about like \(\log n\)).

Exam focus

See "What the exam asks" above for the full list; the one task not worked elsewhere on the page follows.

?Self-test

0The big picture

There is nothing mysterious about an infinite sum. It is simply the limit of a sequence, namely the sequence of partial sums \(s_n=a_1+\dots+a_n\). So everything from Chapter 2 applies. The chapter in four steps:

1. 3.1 Definition and the three model series you must know by heart: geometric (converges iff \(|q|<1\)), harmonic \(\sum\frac1k\) (diverges), telescoping (everything cancels).
2. 3.2 Tests that decide convergence without computing the sum: necessary condition \(a_k\to0\), comparison (majorant/minorant), asymptotic comparison, ratio test, Leibniz test for alternating series.
3. 3.3 Rules: sums of series behave well; rearranging terms is only safe for absolutely convergent series; products via the Cauchy product.
4. 3.4 The exponential series \(\exp(z)=\sum\frac{z^k}{k!}\) and its functional equation \(e^{w+z}=e^we^z\).

3.1Definition and examples [Definition und Beispiele]

Why an infinite sum needs a definition at all. You cannot actually perform infinitely many additions. Picture walking towards a wall 1 m away, each step covering half of the remaining distance: \(\frac12+\frac14+\frac18+\dots\). After \(n\) steps you have walked \(s_n=1-\frac1{2^n}\) metres. You never "finish", but the numbers \(s_n\) approach 1, so it is natural to say the infinite sum is 1. That is exactly the definition: add finitely many terms, then take the limit.

The harmonic series [harmonische Reihe]

The geometric series [geometrische Reihe]

Table from Skript 3.2: the closer \(q\) is to 1, the larger the sum. Watch the starting index: \(\sum_{k=0}^\infty\frac1{2^k}=2\), but \(\sum_{k=1}^\infty\frac1{2^k}=1\). In general \(\sum_{k=m}^\infty q^k=\frac{q^m}{1-q}\) ("first term over one minus ratio"), because \(q^m+q^{m+1}+\dots=q^m(1+q+q^2+\dots)\) Extra.

A telescoping series [Teleskopreihe]

What you must know for 3.1: the definition of convergent/divergent series; the harmonic and geometric series and their behaviour; the value of the geometric series.

3.2Convergence tests [Konvergenzkriterien]

Only the tail matters. Leaving out finitely many terms changes neither convergence of a sequence nor its limit. For a series, dropping terms does not change whether it converges, but it does change the value: \(\sum_{k=0}^\infty2^{-k}=2\) but \(\sum_{k=1}^\infty2^{-k}=1\). So in convergence questions you may always say "for \(k\ge k_0\)".

Extra Why: for \(n\ge k_0\), \(s_n=\underbrace{(a_1+\dots+a_{k_0-1})}_{\text{a fixed number }C}+\sum_{k=k_0}^na_k\). A sequence plus a constant converges exactly when the sequence does (sum rule), and the limit is shifted by \(C\).

The necessary condition

Comparison: majorant and minorant

The ratio test [Quotientenkriterium]

Alternating series: the Leibniz test [Leibnizkriterium]

The script states Leibniz with the sum starting at \(k=0\) and the first term positive. Other starts or signs, like \(\sum_{k\ge1}\frac{(-1)^{k+1}}k\), are covered too: shifting the index or multiplying the whole series by \(-1\) does not change convergence Extra.

What you must know for 3.2: \(a_k\to0\) is necessary but not sufficient; majorant, minorant and ratio tests; the exponential series, the Leibniz series and \(\sum\frac1{k^2}\) and how they behave; deciding convergence of series.

3.3Rules for series [Rechenregeln]

Rearranging terms [Umordnung]

Multiplying series: the Cauchy product

Multiplying two finite sums produces all products \(a_lb_m\). Group them by index sum \(l+m=k\) (like collecting powers of \(x\) when multiplying polynomials):

What you must know for 3.3: only absolutely convergent series may be rearranged arbitrarily; the alternating series; the Cauchy product and how to use it.

3.4The exponential function [Die Exponentialfunktion]

One can show that \(\exp(1)=e=\lim(1+\frac1n)^n\) (Chapter 6 proves the more general \(\lim(1+\frac xn)^n=e^x\)). With the functional equation, \(\exp(k)=e^k\) for all integers \(k\), e.g. \(\exp(2)=e\cdot e\) and \(\exp(-1)=\frac1e\). That is why we write \(e^z\) for \(\exp(z)\), and the functional equation reads \(e^{w+z}=e^we^z\).

What you must know for 3.4: the functional equation; \(\exp(1)=e\).

Mixed exercises from the script [Kapitel 13]

Three convergence questions from the script’s exercise pool — exactly the format of the "does it converge?" exam task.

!What the exam asks from Chapter 3

Series appear in every one of the nine exams — usually two or three short "converges?" items in Task 1 (often answer-only, 1–2 points each) plus sometimes a proof task. The task types:

Drill: which test decides it?

The most common exam task, as a quick drill. For each series, pick the argument that settles it; the explanation gives the full answer you would write.

ΣSummary sheet

1. A series is the sequence of its partial sums. Finitely many terms change the value, never the convergence.
2. Convergent ⇒ terms → 0 (not conversely). Absolutely convergent ⇒ convergent (not conversely).
3. Comparison: \(|a_k|\le b_k\), \(\sum b_k<\infty\) ⇒ convergent; \(a_k\ge b_k\ge0\), \(\sum b_k=\infty\) ⇒ divergent. Asymptotic version: \(a_k\simeq b_k\) (positive) ⇒ same behaviour.
4. Ratio test: limit \(q<1\) conv., \(q>1\) div., \(q=1\) no information.
5. Leibniz: alternating + decreasing + → 0 ⇒ converges; error ≤ first omitted term.
6. Rearranging and multiplying (Cauchy product) are safe for absolutely convergent series.
7. \(e^{w+z}=e^we^z\); \(e^x>0\), strictly increasing; beats every polynomial.

?Self-test

0The big picture

1. 4.1 \(f\) is continuous at \(x\) if \(x_n\to x\) always implies \(f(x_n)\to f(x)\): "\(f\) commutes with limits". Sums, products, quotients and compositions of continuous functions are continuous, so almost everything built from polynomials and exp is continuous.
2. 4.2 Intermediate value theorem: a continuous function on an interval cannot skip values. It gives zeros of equations and the existence of \(\sqrt[n]y\).
3. 4.3 Subsequences and accumulation points. Bolzano–Weierstrass: every bounded sequence has a convergent subsequence (found by repeated halving).
4. 4.4 Compact = closed + bounded. A continuous function on a compact set attains its maximum and minimum. This is why optimisation problems on \([a,b]\) always have a solution.

4.1Definition and simple properties [Definition und einfache Eigenschaften]

The idea. A function is continuous at \(x\) if nearby inputs give nearby outputs: if you approach \(x\), the values approach \(f(x)\). "Approach" is exactly what sequences are good for, so the script defines continuity by testing \(f\) with every sequence that runs into \(x\). Informally: you can draw the graph near \(x\) without lifting the pen.

Two things are packed into this: the limit of \(f(x_n)\) must exist (for every approaching sequence, and always be the same), and it must equal the value \(f(x)\).

What you must know for 4.1: the notion of continuity (at a point); deciding whether a given function is continuous; knowing that exp, polynomials, rational functions and compositions are continuous on their domain.

4.2The intermediate value theorem [Zwischenwertsatz]

Application: n-th roots

What you must know for 4.2: the intermediate value theorem and how to apply it.

4.3Accumulation points [Häufungspunkte]

Bolzano–Weierstrass

What you must know for 4.3: the procedure for finding a convergent subsequence from the proof of Bolzano–Weierstrass.

4.4Existence of maxima and minima

What you must know for 4.4: minimum, maximum, minimum and maximum points; a continuous function on a compact set attains its max and min; which subsets of \(\mathbb R^d\) are compact.

!What the exam asks from Chapter 4

Chapter 4 appears in six of the nine exams, as one of four task types. All of them are worked on this page:

Drill: continuity, IVT, accumulation points, compactness

Ten short items covering the four exam task types. Decide first, then read the explanation — it is the sentence you would write.

ΣSummary sheet

1. Continuity at \(x\): \(x_n\to x\Rightarrow f(x_n)\to f(x)\) for every sequence. Discontinuity: one bad sequence suffices. Equivalent: ε–δ.
2. Polynomials, rational functions, exp, √, |·| and all their sums, products, quotients and compositions are continuous on their domains. Exam work happens only at gluing points / zeros of denominators: continuous there iff value = limit.
3. IVT: continuous on \([a,b]\) + sign change ⇒ zero. Add strict monotonicity for uniqueness. Consequences: fixed points, odd-degree polynomials have roots, \(\sqrt[n]y\) exists.
4. Accumulation point: limit of a subsequence ⇔ infinitely many terms in every ε-neighbourhood. Convergent ⇒ exactly one.
5. Bolzano–Weierstrass: bounded ⇒ convergent subsequence (halve, keep the half with infinitely many terms).
6. Compact = closed + bounded. Continuous on compact ⇒ max and min are attained. Open ends or unbounded domains or discontinuities can destroy this.

?Self-test

0The big picture

1. 5.1 A continuous, strictly increasing function on an interval can be undone. Its inverse is again continuous and increasing, and its graph is the mirror image at the diagonal \(y=x\).
2. 5.2 \(\ln\) = inverse of \(\exp\). The functional equation \(e^{a+b}=e^ae^b\) turns into \(\ln(xy)=\ln x+\ln y\). Then \(x^a:=e^{a\ln x}\) defines powers with any real exponent. Growth: \(e^x\) beats every power, every power beats every power of \(\ln\).
3. 5.3 Plug imaginary numbers into exp: \(e^{ix}\) lies on the unit circle, and its coordinates define \(\cos x\) and \(\sin x\). Trig identities follow from \(e^{i(x+y)}=e^{ix}e^{iy}\). Key limits: \(\frac{\sin x}x\to1\), \(\frac{1-\cos x}{x^2/2}\to1\).

5.1Inverse functions [Umkehrfunktion]

What you must know for 5.1: the definition of the inverse function; determining the inverse in concrete cases.

5.2The logarithm [Logarithmusfunktion]

Growth rates

General powers and logarithms to base b

What you must know for 5.2: the rules for the logarithm; the growth behaviour of log and exp; the functions \(x^a\), \(b^x\) and \(\log_bx\).

5.3Trigonometric functions [Trigonometrische Funktionen]

What you must know for 5.3: Euler's formula and deriving trig identities from it; the definitions of sin, cos, tan, cot and their inverses; the asymptotic behaviour of \(\sin x\) and \(1-\cos x\) for \(x\to0\).

!What the exam asks from Chapter 5

Chapter 5 appears in six of the nine exams — WiSe19, WiSe20, WiSe22, WiSe23, SoSe23, WiSe24 — mostly as short answer-only items in Task 1:

And as a first step inside later tasks. Chapter 5 is also what makes several non-Chapter-5 exam tasks start: SoSe21 4b (extrema of \(x^x\) on \((0,\infty)\)) only becomes differentiable-looking after writing \(x^x=e^{x\ln x}\); WiSe24 4a differentiates \(\log\frac1{1+e^x}\); SoSe21 1a asks for the accumulation points of \(e^{i\pi n/2}(1-e^{i\pi n})\), i.e. Euler's formula on the unit circle; WiSe19 1c (\(n^7\sin 3^{-n}\to0\)) needs \(|\sin u|\le|u|\) plus "exponential beats power". If a task looks unfamiliar, check first whether a Chapter 5 rewrite turns it into something standard.

Drill: which term dominates?

Answer-only "≃" and limit items are quick points once the growth ladder is automatic. Pick the answer; the explanation shows the reasoning to write down.

ΣSummary sheet

?Self-test

0The big picture

1. 6.1 \(f=O(g)\): \(f\) is at most a constant times \(g\) (near \(x_0\)). \(f=o(g)\): \(f\) is negligible compared to \(g\) (\(\frac fg\to0\)).
2. 6.2 \(f\) is differentiable at \(x_0\) if \(f(x)=f(x_0)+f'(x_0)(x-x_0)+o(|x-x_0|)\): the tangent line has an error that is small even compared to the distance \(|x-x_0|\). Equivalently, the difference quotient converges. Differentiable ⇒ continuous, not conversely (\(|x|\)).
3. 6.3 Rules: sum, product, quotient, chain rule, derivative of the inverse. With them, \((x^n)'=nx^{n-1}\), \((a^x)'=a^x\ln a\), \((\ln x)'=\frac1x\), \((\tan)'=1+\tan^2\), \((\arctan)'=\frac1{1+x^2}\), …

6.1Landau symbols [Landau-Symbole]

What you must know for 6.1: working with o and O.

6.2Differentiability [Differenzierbarkeit]

Goal: approximate \(f\) near \(x_0\) by a straight line that describes how \(f\) changes there.

What you must know for 6.2: the definitions of derivative and tangent; the derivatives of exp, sin and cos.

6.3Rules of differentiation [Ableitungsregeln]

What you must know for 6.3: the sum, product, quotient and chain rules and the rule for the inverse function; the derivatives of important functions, in particular \(x^n,a^x,x^a,\tan,\arctan\).

Mixed exercises from the script [Kapitel 13]

Three true/false statements about \(o\) and \(O\) from the script’s exercise pool. The method is always the same: translate the symbol into a limit, then compute the limit.

!What the exam asks from Chapter 6

Chapter 6 appears in six of the nine exams. Landau items are answer-only and quick; derivatives show up everywhere later (curve sketching, Taylor, ODEs), so fluency matters more than any single task.

Extra Beyond these three types, a chapter-6 step is buried inside many later tasks, and that is where most of the marks for this chapter actually are: every curve sketch needs \(f'\) and \(f''\) (WiSe23 A3, WiSe18 A1, WiSe20 A4, §7.7); SoSe21 4b needs the rewrite \(x^x=e^{x\ln x}\) before it can be differentiated at all; l’Hospital (WiSe18 1a, WiSe24 1c) differentiates numerator and denominator separately; WiSe22 4b differentiates an integral with the fundamental theorem; and the several-variable tasks (WiSe19 1g, WiSe24 5a, SoSe23 A3) are these same rules applied one variable at a time.

Drill: derivatives and Landau

Ten quick items in exam style. Do them on paper first, then check.

ΣSummary sheet

Domains: \(x^a\) and \(\ln x\) on \((0,\infty)\), \(a^x\) for \(a>0\) on all of \(\mathbb R\), \(\arcsin\) on \((-1,1)\), \(\tan\) away from \(\frac\pi2+k\pi\). The \(\ln|x|\) row is not in Chapter 6 — the script only does \(\ln\) on \((0,\infty)\) (Beispiel 6.23) — but it is the antiderivative you need in Chapter 8, and it follows from the chain rule: for \(x<0\), \((\ln(-x))'=\frac{-1}{-x}=\frac1x\).

1. \(f=O(g)\): \(|f|\le C|g|\) near \(x_0\). \(f=o(g)\): \(f/g\to0\). Check via \(\lim|f/g|\). Near 0 the lowest power dominates, near ∞ the highest.
2. \(f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}h\iff f(x)=f(x_0)+f'(x_0)(x-x_0)+o(|x-x_0|)\).
3. Differentiable ⇒ continuous. \(|x|\) at 0: continuous, not differentiable. \(x^2\sin\frac1x\): differentiable, \(f'\) not continuous.
4. At gluing points of piecewise functions, use the definition (difference quotient), not the rules.
5. Rules: \((fg)'=f'g+fg'\), \(\left(\frac fg\right)'=\frac{f'g-fg'}{g^2}\), \((f\circ g)'=f'(g)\,g'\), \((f^{-1})'=\frac1{f'(f^{-1})}\).

?Self-test

0The big picture

1. 7.1 At an interior local extremum, \(f'=0\) (necessary, not sufficient).
2. 7.2 Mean value theorem: the average slope \(\frac{f(b)-f(a)}{b-a}\) is attained as a derivative \(f'(\xi)\) somewhere in between. This is the bridge from "derivative info" to "function info".
3. 7.3 Consequences: \(f'>0\) ⇒ strictly increasing; \(f'=0\) ⇒ constant; sign change of \(f'\) ⇒ extremum; inequalities like \(1+x\le e^x\).
4. 7.4 l’Hospital: for "\(\frac00\)" or "\(\frac\infty\infty\)", replace \(\frac fg\) by \(\frac{f'}{g'}\).
5. 7.5–7.6 Higher derivatives; \(f''\ge0\) ⇔ convex; \(f'(x_0)=0\) and \(f''(x_0)>0\) ⇒ local minimum.
6. 7.7 Curve sketching puts it all together.

7.1Extrema [Extrema]

What you must know for 7.1: local and global min/max; \(f'(x)=0\) is necessary for a local extremum of a differentiable function; finding local and global extrema.

7.2The mean value theorem [Mittelwertsatz]

What you must know for 7.2: the mean value theorem, the generalised MVT and Rolle's theorem, and how to apply them.

7.3Monotonicity [Monotonie]

What you must know for 7.3: the definition of monotonicity and the monotonicity criterion; determining monotonicity and extrema of a given function.

7.4Computing limits: l’Hospital [Regel von l’Hospital]

What you must know for 7.4: the rule of l’Hospital and how to apply it.

7.5Higher derivatives [Höhere Ableitungen]

What you must know for 7.5: computing higher derivatives.

7.6Convexity [konvex / konkav] — the script calls the section Krümmungsverhalten, curvature behaviour

What you must know for 7.6: the sufficient criterion for local min/max; determining the curvature behaviour of a function.

7.7Curve sketching [Kurvendiskussion]

What you must know for 7.7: for a given function, determining boundary behaviour, derivatives, extrema, monotonicity and curvature.

!What the exam asks from Chapter 7

Chapter 7 is in every exam, usually with the biggest single task (a curve sketch worth 8–11 points):

Drill: classify points and compute limits

ΣSummary sheet

1. Interior local extremum ⇒ \(f'=0\). Not conversely (\(x^3\)); endpoints separately.
2. MVT: \(\frac{f(b)-f(a)}{b-a}=f'(\xi)\). Rolle: equal ends ⇒ \(f'(\xi)=0\). Generalised: \(\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f'(\xi)}{g'(\xi)}\).
3. \(f'>0\Rightarrow\) strictly increasing; \(f'\ge0\iff\) increasing; \(f'=0\iff\) constant (on intervals).
4. Extremum tests: sign change of \(f'\) (always works), or \(f''(x_0)\neq0\) (fails when \(f''(x_0)=0\)).
5. \(1+x\le e^x\), \(\ln(1+x)\le x\). Prove inequalities by minimising the difference.
6. l’Hospital only for \(\frac00,\frac\infty\infty\); rewrite other forms first; often an \(o\)-expansion is faster.
7. \(f''\ge0\iff\) convex (graph below chords).

?Self-test

0The big picture

1. 8.1 Chop \([a,b]\) into small pieces, add up rectangles \(f(\xi_j)\cdot\text{width}\) (Riemann sums), let the pieces shrink. The limit is \(\int_a^bf(x)\,dx\).
2. 8.2 The integral is linear, monotone, additive over intervals; every continuous function is integrable; mean value theorem of integration.
3. 8.3 Fundamental theorem: \(F(x)=\int_a^xf(t)\,dt\) has \(F'=f\). Consequence: \(\int_a^bf=F(b)-F(a)\) for any antiderivative \(F\).
4. 8.4 Rules: \(\int fg'=[fg]-\int f'g\) (by parts); \(\int f(g(t))g'(t)\,dt=\int f(x)\,dx\) (substitution).

8.1Definition of the integral

For a constant function \(f=c\) on \([a,b]\), the area is \(c(b-a)\). For other functions, approximate by rectangles:

Try "smallest value" and "largest value" in the widget: they aim at the lower and upper sum, the smallest and largest Riemann sum a given partition allows. (The widget finds each strip's smallest and largest value by sampling 25 points in it, so for a wiggly \(f\) it is very close rather than exact.) For a continuous \(f\) the two squeeze together as \(n\) grows — and every other choice of \(\xi_j\) lies between them, which is why all choices share the same limit.

What you must know for 8.1: the definition of the integral.

8.2Important properties

What you must know for 8.2: applying the properties of the integral.

8.3Antiderivatives [Stammfunktionen]

The first eight entries are the script's own table (after Satz 8.9). The last two are added: \(\int\ln x\,dx\) is Beispiel 8.17 further down this page, and \(\int a^x\,dx=\frac{a^x}{\ln a}\) comes from the solution to Blatt 11.2(d) — both are standard exam fare, which is why they sit here rather than only in the text.

Skript 8.13: the simplest use of the table — term by term (linearity), then \(F(1)-F(0)\): \[\int_0^1(x^7+5x^3+2)\,dx=\left[\frac{x^8}8+\frac54x^4+2x\right]_0^1=\frac18+\frac54+2=\frac{27}8.\]

Why \(\ln|x|\): for \(x<0\), \((\ln(-x))'=\frac{-1}{-x}=\frac1x\). Always check an antiderivative by differentiating it. That takes seconds.

What you must know for 8.3: finding an antiderivative; the fundamental theorem (important!); computing integrals with an antiderivative; partial fractions.

8.4Integration rules [Integrationsregeln]

Integration by parts [Partielle Integration]

Substitution [Substitutionsregel]

What you must know for 8.4: integration by parts and the substitution rule.

Mixed exercises from the script [Kapitel 13]

Three integrals from the script’s exercise pool, one per technique.

!What the exam asks from Chapter 8

Every exam has at least one integral, often a proof about integrals as well:

Drill: which technique?

Deciding the technique is half the task. For each integral, pick the method (and check the answer in the explanation).

ΣSummary sheet

1. \(\int_a^bf=\lim\) of Riemann sums \(\sum f(\xi_j)\Delta x_j\); continuous ⇒ integrable; signed area.
2. Linear, monotone, additive; \(|\int f|\le\int|f|\); \(\int_a^bf=f(\xi)(b-a)\) for continuous \(f\).
3. FTC: \(\frac{d}{dx}\int_a^xf(t)\,dt=f(x)\), and \(\int_a^bf=F(b)-F(a)\). Use it to differentiate integral-defined functions without computing them.
4. By parts: \(\int fg'=[fg]-\int f'g\); choose \(f\) to simplify; twice + solve for cyclic cases.
5. Substitution: \(x=g(t)\), \(dx=g'(t)dt\), change the limits. Look for "function and its derivative".
6. Partial fractions for rational functions; odd functions on symmetric intervals integrate to 0.
7. Always check antiderivatives by differentiating.

?Self-test

0The big picture

1. 9.1 \(\int_a^\infty f=\lim_{M\to\infty}\int_a^Mf\): an improper integral is a limit of ordinary integrals, just like a series is a limit of partial sums. Benchmark: \(\int_1^\infty x^{-\alpha}\) converges iff \(\alpha>1\); \(\int_0^1x^{-\alpha}\) iff \(\alpha<1\).
2. 9.2 For continuous integrands you may swap integration order (Fubini) and differentiate under the integral sign.
3. 9.3 \(\int\sum=\sum\int\) if there is a convergent majorant. Gives \(\ln(1+x)=\sum(-1)^k\frac{x^{k+1}}{k+1}\) and \(\ln2=1-\frac12+\frac13-\dots\)
4. 9.4 For monotone \(f\), \(\sum f(k)\) is squeezed between integrals. Integral test: \(\sum f(k)\) and \(\int f\) converge together.

9.1Improper integrals [Uneigentliche Integrale]

What you must know for 9.1: recognising improper integrals; for which α the integrals \(\int_0^1x^{-\alpha}\) and \(\int_1^\infty x^{-\alpha}\) converge or diverge.

9.2Parameter-dependent integrals [Parameterabhängige Integrale]

For \(f(x,y)\) and fixed \(x\), the derivative of \(y\mapsto f(x,y)\) is the partial derivative \(\frac{\partial f}{\partial y}=\partial_yf=\partial_2f\) (differentiate in \(y\), treat \(x\) as a constant; Chapter 11 in detail).

What you must know for 9.2: computing partial derivatives; Satz 9.8.

9.3Swapping summation and integration

What you must know for 9.3: the sufficient criterion for swapping sum and integral; the power series of \(\ln(1+x)\) and \(\frac1{1-x}\).

9.4Estimating sums and series [Abschätzungen von Summen und Reihen]

What you must know for 9.4: estimating sums with integrals; the integral test for series.

Mixed exercises from the script [Kapitel 13]

!What the exam asks from Chapter 9

Chapter 9 appears in six of the nine exams, most heavily in WiSe24 (four subtasks):

Drill: does it converge?

ΣSummary sheet

1. Improper integral = limit of proper integrals; split at an interior point if both ends are critical; each part must converge separately.
2. \(\int_1^\infty x^{-\alpha}<\infty\iff\alpha>1\); \(\int_0^1x^{-\alpha}<\infty\iff\alpha<1\). Compare other integrands with these.
3. Continuous \(f(x,y)\): Fubini; \(\frac d{dy}\int f\,dx=\int\partial_yf\,dx\).
4. \(\int\sum=\sum\int\) with a convergent majorant; otherwise it can fail (spikes). \(\ln(1+x)=\sum(-1)^k\frac{x^{k+1}}{k+1}\) on \((-1,1]\).
5. Monotone \(f\): \(\sum f(k)\) is sandwiched by integrals. \(H_n\simeq\ln n\), \(\sum k^\alpha\simeq\frac{n^{\alpha+1}}{\alpha+1}\), \(\ln n!\simeq n\ln n\).
6. Integral test: for decreasing \(f\ge0\), \(\sum f(k)\) and \(\int_1^\infty f\) converge together.

?Self-test

0The big picture

1. Known power series so far: \(e^x=\sum\frac{x^k}{k!}\) (all \(x\)) and \(\ln(1+x)=\sum(-1)^k\frac{x^{k+1}}{k+1}\) — these two are the script's own list at the head of the chapter — plus the geometric series \(\frac1{1-x}=\sum x^k\) (\(|x|<1\)) from Chapter 3.
2. Why they matter: truncating gives polynomial approximations (numerics); conversely, a closed formula for \(\sum a_kx^k\) (a generating function) tells you things about the \(a_k\) (combinatorics, probability).
3. 10.1 Taylor's formula: \(f(a+h)=\sum_{k\le n}\frac{f^{(k)}(a)}{k!}h^k+\text{remainder}\). Taylor series = limit, when the remainder → 0.
4. 10.2 Every power series has a radius \(r\): absolutely convergent for \(|x|<r\), divergent for \(|x|>r\). Inside, differentiate/integrate term by term.

10.1Taylor's formula [Taylorsche Formel]

Where the coefficients come from. Suppose \(f(a+h)=f(a)+f'(a)h+a_2h^2+\dots+a_nh^n+o(h^n)\). Then each \(a_k\) is a limit of the form \(\frac{f(a+h)-(\dots)}{h^k}\), which is "\(\frac00\)". Applying l’Hospital \(k\) times gives \(a_k=\frac{f^{(k)}(a)}{k!}\). This matches Lemma 7.22 for polynomials: the coefficients are the derivatives at the centre, divided by factorials.

What you must know for 10.1: Taylor's formula including the Cauchy and Lagrange remainders; when a function can be expanded into its Taylor series; the binomial series.

10.2Radius of convergence and analytic functions

What you must know for 10.2: what the radius of convergence is and how to determine it; the derivative, an antiderivative and the coefficients of a power series.

!What the exam asks from Chapter 10

Chapter 10 appears in five of the nine exams, almost always as short answer-only items:

Drill: Taylor polynomials, values, radii

ΣSummary sheet

1. \(T_{n,a}f(x)=\sum_{k\le n}\frac{f^{(k)}(a)}{k!}(x-a)^k\); error \(o((x-a)^n)\), or exactly \(\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-a)^{n+1}\) (Lagrange).
2. To get Taylor polynomials of composite functions, substitute into known series (uniqueness makes it legitimate).
3. Radius via ratio or root test; boundary points by hand.
4. Inside the radius: differentiate/integrate term by term.

?Self-test

0The big picture

Everything is the one-variable theory with vectors and matrices in place of numbers:

Roadmap. §11.1 treats functions out of one variable into \(\mathbb R^n\) (curves): differentiate each component. §11.2 treats functions from \(\mathbb R^n\) into \(\mathbb R\): differentiate in one variable at a time, and Lemma 11.7 says these partial derivatives together give the best linear approximation. §11.3 goes to second order (Hessian) — which is exactly what decides extrema, just as \(f''\) did in Chapter 7 — and ends with maps \(\mathbb R^n\to\mathbb R^m\) (Jacobian), where everything is combined.

The linear algebra you need here Extra

The script assumes matrices, eigenvalues and definiteness are known. Here is everything this chapter actually uses, from scratch. If it is familiar, skip ahead.

11.1Curves [Kurven]

What you must know for 11.1: what a curve is; determining its tangent vector.

11.2Partial derivatives [Partielle Ableitungen]

What you must know for 11.2: for \(f:\mathbb R^n\to\mathbb R\), the partial derivatives, grad f, Df and tangent planes; the notion of total differentiability.

11.3Higher derivatives and extrema

Vector-valued functions: the Jacobian matrix