Title: HS25 / Analysis I / Vorlesung
Tags: Definitions Theorems Propositions Lemmas Corollary

Q: Gruppe
A: Associativity a*(b*c) = (a*b)*c
Neutral Element e
Innverse element a^-1

Q: Commutative / Abelian Group
A: Group with a*b = b*a

Q: Ring
A: Set R with operations +, * such that
(G, +) is abelian group
* is associative a*(b*c)=(a*b)*c
* has a neutral element
* distributes over + a*(b+c) = a*b + a*c

Q: Commutative Ring
A: Ring where * is commutative a*b = b*a

Q: Field
A: Commutative Ring R s.t. every element other than zero has a * inverse

Q: Relation
A: X set, \(R\subseteq X\times X\)

Q: reflexive Relation
A: x ~ x

Q: transitive relation
A: x ~ y and y ~ z => x ~ z

Q: symmetric relation
A: x ~ y => y ~ x

Q: antisymmetric relation
A: x ~ y and y ~ x => x=y

Q: Equivalence Relation
A: Relation that is reflexive, transitive and symmetric

Q: Order Relation
A: Relation that is reflexive, transitive and antisymmetric

Q: Ordered Field
A: Field F, with Order Relation \(\ge\)
- \(x,y\in F \implies x\le y \,\vee\, y\le x\)
- \(x,y,z\in F: x\le y \implies x+z\le y+z\)
- \(x,y\in F: 0\le x \,\wedge\, 0\le y \implies 0\le x\cdot y \)

Q: Function
A: Given sets \(X, Y\): Function assigns an element of \(Y\) to every element of \(X\)

Q: Modulus Function
A: \(|\cdot |: F\mapsto F\) defined by
\[ |x| = \begin{cases} x & \text{if } x\ge 0\\ -x & \text{if } x<0 \end{cases} . \]

Q: Sign Function
A: \(sgn: F \mapsto \{-1,0,1\}\) defined by
            \[sgn(x) = \begin{cases} 1 & \text{if } x>0\\ 0 & \text{if } x=0\\ -1 & \text{if } x<0 \end{cases} .\]

Q: Completeness Axiom
A: \(X,Y\subseteq F\setminus\emptyset\). Assume \(x\le y\forall x\in X,y\in Y\). Then \(\exists c\in F\) s.t.
\[x\le c\le y \forall x,y\]

Q: Real Numbers
A: Any complete, ordered field

Q: Injective Funtion
A: \[x_1\neq x_2 \implies f(x_1)\neq f(x_2)\]

Q: Surjective function
A: \[\forall y\in Y\,\exists x\in X : f(x)=y\]

Q: Bijective function
A: Function that is injective and surjective

Q: Image of \(A\) through \(f\)
A: \[f(A)=\{y\in Y:\exists x\in A : y=f(x)\}\]

Q: Preimage of \(A\) through \(f\)
A: \(A\subseteq Y\)
\[f^{-1}(A)=\{x\in X:f(x)\in A\}\]

Q: closed interval
A: \[[a,b]=\{x\in\mathbb{R}\mid a\le x\le b\}\]

Q: open interval
A: \[(a,b)=\{x\in\mathbb{R}\mid a< x< b\}\]

Q: unbounded interval
A: Interval including \(\pm\infty\)

Q: Neighborhood
A: \(x\in\mathbb{R}\). Set that contains open interval \(I\), s.t. \(x\in I\)

Q: Delta-Neighborhood
A: Given \(\delta>0\), the open interval \((x-\delta, x+\delta)\)

Q: Open Set
A: Set \(U\in\mathbb{R}\) is open if \(\forall x\in U\exists\) open interval I s.t. \(x\in I\subseteq U\).

Q: Closed set
A: \(V\)is closed if \(\mathbb{R}\setminus V\) is open.

Q: Complex conjugate
A: For \(z=x+iy\): \(\overline{z}=x-iy\)

Q: Modulus of complex numbers
A: \[|z|=\sqrt{z\cdot\overline z}=\sqrt{x^2+y^2}\]

Q: Triangle inequality for \(\mathbb{C}\)
A: \[|z+w|\le|z|+|w|\]

Q: Cauchy Schwarz Inequality
A: Given \(z,w \in \mathbb{C}\)
\[x_1x_2+y_1y_2\le|z||w|\]

Q: Open Disk
A: Given \(z\in\mathbb{C}\) and \(r>0\)
\[ B(z,r)=\{w\in\mathbb C \mid |w-z|<r\} \]

Q: Closed Disk
A: Given \(z\in\mathbb{C}\) and \(r>0\)
\[\overline{B(z,r)}=\{w\in\mathbb C \mid |w-z|\le r\}\]

Q: Open Set (on \(\mathbb{C}\))
A: Set \(U\) such that\(\forall z\in U \exists r>0\) such that \(B(z,r)\subseteq U\)

Q: Closed Set (on \(\mathbb{C}\))
A: Set \(C\) s.t. \(\mathbb{C}\setminus C\) is open

Q: Bounded Above
A: \(\exists s\in\mathbb R\) such that \(x\le s\forall x\in X\)

Q: Maximum
A: Upper Bound \(s\) that belongs to \(X\)

Q: Bounded set
A: Set bounded above and below

Q: Supremum
A: \[\min(\{a\in \mathbb R \mid a\ge x\,\forall x\in X\} )\]
The lowest upper bound

Q: Extended Real Line
A: \[\overline{\mathbb{R}}=\mathbb R \cup \{\infty, -\infty\}\]

Q: Archimedian Principle
A: Given \(x\in\mathbb{R}\,\exists!n\in\mathbb Z\) such that \(n\le x < n+1\)

Q: Integer Part of \(x\)
A: \(\lfloor x\rfloor\) is the \(n:n\le x < n+1\)

Q: Fractional Part of \(x\)
A: \(\{x\}=x-\lfloor x\rfloor\)

Q: Dense set
A: A set \(X\) is dense in \(\mathbb{R} \)if every open interval contains an element of \(X\)

Q: Power Set
A: \[\mathcal{P}(X)=\{Y|Y\subseteq X\}\]

Q: Countable Set
A: \[\exists f:X\to\mathbb{N} \text{ bijective}\]

Q: Uncountable Set
A: \(X\) is infinite and not countable

Q: Sequence
A: Function from \(\mathbb{N}\to\mathbb{R}\) where \(a_n\) is the n-th element of the sequence

Q: Constant sequence
A: \[x_i=x_j\,\forall i,j\]

Q: Eventually Constant Sequence
A: \[\exists N\in\mathbb N:x_i=x_j\forall i,j\ge N\]

Q: Converging Sequence
A: \[\exists A\in\mathbb{R}:\forall\varepsilon>0,\exists N\in\mathbb{N}:\forall n\ge N, |x_n-A|<\varepsilon\]

Q: Subsequence
A: \[\left(x_{n_k}\right)_{k\ge0}\]

Q: Accumulation point
A: \((x_n)_{n=0}^\infty\). We say \(A\in\mathbb{R}\) is an accumulation point if\[\forall \varepsilon > 0, \forall N\in\mathbb{N}\,\exists n\ge N: |x_n - A| < \varepsilon
.\]

Q: Limit computations
A: Let \((x_n)\) and \((y_n)\) converge to \(A,B \) respectively then
- \(\lim (x_n+y_n)=A+B\)
- \(\lim (x_n\cdot y_n)=A\cdot B\)
- \(\lim(c\cdot x_n) = c\cdot A\)
- \(\lim(x_n^{-1})=A^{-1} \) if \(x_n\neq 0\) and \(A\neq0\)

Q: Limits with inequalities
A: \((x_n),(y_n)\) with limits \(A,B\)
- If \(A<B\) then \(\exists N:x_n<y_n\forall n\ge N\)
- If \(\exists N:x_n\le y_n\forall n\ge N\), then \(A\le B\)

Q: Sandwich Lemma
A: \((x_n),(y_n),(z_n)\) such that \(\exists N\)
\[x_n\le y_n\le z_n \forall n\ge N\]If \(x_n\) and \(z_n\) converge to the same limit \(A\) then \(y_n\) does so too

Q: Bounded Sequence
A: \[\exists M>0:|x_n|\le M\forall n\in \mathbb{N}\]

Q: Monotone Sequences
A: Increasing: \(m>n \implies x_m \ge x_n\)
strictly increasing: \(m>n \implies x_m>x_n\)
Decreasing: \(m>n \implies x_m\le x_n\)
Strictly decreasing: \(m>n \implies x_m<x_n\)

Q: Thm: Bounded, monotone sequences ...
A: Converge

Q: Thm: Monotone, Convergent Series ...
A: Are bounded

Q: Tail of a sequence
A: Givne \((x_n)\), and \(N\in\mathbb N\) then the tail is\[X_{\ge N}=\{x_N, x_{N+1},\ldots\}\]

Q: Superior Limit
A: \[ \limsup_{n\to\infty} x_n \coloneqq \lim_{n\to\infty} \underbrace{\left(\sup_{k\ge n} x_k\right)}_{=s_n}\]

Q: Inferior Limit
A: \[ \liminf_{n\to\infty} x_n \coloneqq \lim_{n\to\infty} \underbrace{\left(\inf_{k\ge n} x_k\right)}_{=i_n}\]

Q: A bounded sequence converges if (limsup /liminf)
A: \[\liminf_{n\to\infty}x_n=\limsup_{n\to\infty}x_n\]

Q: Let \((x_n)_{n\ge 0}\) be a bounded sequence with \(A=\limsup x_n\). Then A is
A: an Accumulation point

Q: Cauchy Sequence
A: \[\forall\varepsilon>0,\,\exists N:|x_n-x_m|<\varepsilon,\quad\forall n,m\ge N\]

Q: All cauchy sequences are
A: bounded

Q: Cauchy sequence is equivalent to
A: Convergent Sequence

Q: Divergence (for + and -)
A: \(+\infty\):
\[\forall M > 0 \,\exists N:x_n>M,\forall n\ge N\]\(-\infty\):
\[\forall M > 0 \,\exists N:x_n<-M,\forall n\ge N\]

Q: Convergence of Complex Sequence
A: if \(z=x+iy\), both \((x_n)\) and \((y_n)\) converge.

Q: Function bounded from above
A: \[\exists M>0:\forall x\in D: f(x)\le M\]

Q: function bounded from below
A: \[\exists M>0:\forall x\in D: f(x)\ge -M\]

Q: Bounded function
A: \[\exists M>0:\forall x\in D: |f(x)|\le M\]

Q: Monotone functions
A: If \(x<y\) then
- increasing if \(f(x)\le f(y)\)
- strictly increasing if \(f(x)<f(y)\)
- decreasing if \(f(x)\ge f(y)\)
- strictly decreasing if \(f(x)>f(y)\)

Q: Continuous function at \(x_0\in D\)
A: \[\forall\varepsilon>0,\exists\delta>0:\forall x\in D, |x-x_0|<\delta\implies |f(x)-f(x_0)|<\varepsilon\]

Q: Restriction of a function
A: Given \(D'\subseteq D\)
\[f|_{D'}(x)=f(x)\,\forall x\in D'\]

Q: Algebra of Continuous functions
A: Given \(f_1, f_2: D\to \mathbb R\)
- \(f_1+f_2\) is continuous
- \(f_1\cdot f_2\) is continuous
- \(af_1\) is continuous for \(a\in\mathbb R\)

Q: Continuity and Composition
A: If \(f,g\) are continuous then \(g\circ f\) is so aswell

Q: Continuity vs Sequential continuity
A: \(f\) is continuous at \(\overline x\) if and only if
\[\forall (x_n)\subseteq D, x_n \to \overline x\implies f(x_n)\to f(\overline x)\]

Q: Intermediate Value Theorem
A: Given \(f:[a,b]\to\mathbb R\) continuous. If \(f(a)<f(b)\), then\[\forall c\in[f(a),f(b)]\,\exists\,\overline{x}\in[a,b]:f(\overline x)=c\]

Q: Inverse Function Theorem
A: Given interval \(I\) and \(f:I\to\mathbb R\) continuous, strictly monotone. Then
- \(f(I)\subseteq\mathbb R\) is an interval
- The inverse is also continuous and strictly monotone.

Q: Compact Interval
A: Bounded, Closed Interval

Q: Maximum and Minimum of function
A: f takes its maximum value at \(x_0\) if \(f(x)\le f(x_0)\quad \forall x\in D\)

Q: If \(f:[a,b]\to\mathbb R\) is continuous, then \(f\) takes
A: its maximum and minimum

Q: Uniform continuity
A: \[\forall\varepsilon >0\,\exists\delta>0:\forall x,y\in D, |x-y|<\delta \implies|f(x)-f(y)|<\varepsilon\]

Q: \(f:[a,b]\to\mathbb R\) continuous, the \(f\) is also
A: uniformly continuous

Q: Bernoulli inequality
A: \[\forall a\in\mathbb R_{\ge -1},n\ge 1, (1+a)^n \ge 1+na\]

Q: Exponential Function
A: \[\exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n\]

Q: Logarithmic Function
A: Inverse of the exponential function

Q: Accumulation Point of a set
A: Given \(D\subseteq \mathbb R, x_0\in\mathbb R\). \(x_0\) is an accumulation point of \(D\) if\[\forall \delta>0, (x_0-\delta,x_0+\delta)\cap D\setminus\{x_0\}\neq\emptyset\]

Q: Limit of a function
A: \[\lim_{x\to x_0}f(x)=\forall\varepsilon>0,\exists\delta>0:|x-x_0|<\delta\implies |f(x)-L|<\varepsilon\]

Q: Calculation Rules and Inequalities for limits of functions
A: Let \(\lim_{x\to x_0} f(x)=L_1\),\(\lim_{x\to x_0}g(x)=L_2\) then
- \(f(x)+g(x)\to L_1 + L_2\)
- \(f(x)\cdot g(x)\to L_1\cdot L_2\)
- \(f\le g\implies L_1 \le L_2\)
- Sandwich lemma

Q: Improper Limit
A: Let \(f:D\to \mathbb R, x_0\) be an accumulation point of \(D\). We say that
        \(f\) diverges to \(+\infty\) as \(x\to x_0\) if
        \[\forall M>0,\exists \delta>0: x\in D, |x-x_0| < \delta \implies f(x) \ge M\]
        In this case, we write
        \[\lim_{x \to x_0} f(x) = +\infty\]

Q: One sided Accumulation point
A: \(x_0\) is an accumulation point from the right if \[\forall \delta >0 (x_0,x_0+\delta)\cap D\neq \emptyset\]

Q: Limit at \(+\infty\)
A: Let \(f:D\to \mathbb R\). assume \(D\cap (R, \infty) \neq \emptyset \forall R>0\).
        We say that \(\lim_{x \to \infty} f(x) = L\) if
        \[\forall \varepsilon > 0, \exists R > 0: x\in D\cap(R, \infty) \implies |f(x) - L| < \varepsilon\]

Q: Jump point
A: If the right and left limit exist, but are different

Q: Landau Big O
A: Given \(f,g:D\to \mathbb R, x_0\) an accumulation point of \(D\).
        We write \(f(x)=O(g(x))\) as \(x\to x_0\) if \(\exists M>0,\delta>0\) such that
        \[|f(x)| \le M|g(x)|, \forall x\in D, |x-x_0| < \delta\]

Q: Landau Little o
A: Given \(f,g:D\to \mathbb R, x_0\) an accumulation point of \(D\).
We write \(f(x)=o(g(x))\) as \(x\to x_0\) if \(\forall \varepsilon > 0, \exists \delta>0\) such that
\[|f(x)| \le \varepsilon |g(x)|, \forall x\in D, |x-x_0| < \delta\]

Q: Sequence of Functions
A: Family of functions indexed by \(\mathbb N\)

Q: Pointwise Convergence
A: \[\forall x \in D, f_n(x)\to_{n\to\infty} f(x)\]

Q: Unifrom convergence for Sequence of Functions
A: \[\forall\varepsilon > 0 \exists N\in \mathbb N:|f_n(x)-f(x)|<\varepsilon,\forall x\in D, \forall n\ge N\]

Q: Series
A: Given sequence \(a_n\), the corresponding sequence is\[\sum_{k=0}^na_k\]

Q: Majorant and Minorant Criterion
A: Let \(0\le a_k \le b_k\). Then
\[0\le\sum_{k=0}^\infty a_k\le\sum_{k=0}^\infty b_k\]

Q: Cauchy Condensation Test
A: Let \(a_k\ge 0\) and \(a_k\) be decreasing. Then
\[\sum_{k=0}^\infty a_k \text{ converges}\iff\sum_{k=0}^\infty 2^ka_{2^k} \text{ converges}\]

Q: Absolute Convergence
A: Series converges absolutely if \(\sum |a_k|\) converges.

Q: Conditional Convergence
A: A series converges conditionally if \(\sum a_k\) converges but \(\sum |a_k| = +\infty\)

Q: Alternating Series
A: \[\sum_{k=0}^\infty(-1)^ka_k\]

Q: Leibniz Criterion
A: If \(a_k\ge 0\) and montonously decreasing to 0, then the alternating series converges

Q: Cauchy Criterion
A: A series converges iff \(\forall \varepsilon >0\,\exists N\) sucht that
\[\left|\sum_{k=m+1}^na_k\right|<\varepsilon \quad \forall n>m\ge N\]

Q: Cauchy Root Criterion
A: Given \((a_k)_{k\ge 0}\), define
\[\alpha = \limsup_{n\to\infty}\sqrt[n]{|a_n|}\]The series converges absolutely if \(\alpha <1\), and does not converge if \(\alpha>1\)

Q: DAlamberts Quotient Criterion
A: Assume \(a_n\neq 0\,\forall n\). Let
\[\alpha = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|\]The series converges absolutely if \(\alpha <1\), and does not converge if \(\alpha>1\)

Q: Cauchy Product Formula
A: If \(\sum_{k=0}^{\infty} a_k\) and \(\sum_{k=0}^{\infty} b_k\) are absolutely convergent series, then
        \[\sum_{k=0}^{\infty} a_k \cdot \sum_{k=0}^{\infty} b_k = \sum_{n=0}^{\infty} \left( \sum_{k=0}^{n} a_{n-k} b_k \right)\]

Q: Convergence of complex Series
A: Both \(\sum x_n\) and \(\sum y_n\) converge

Q: Absolute convergence of complex series
A: \[ \sum_{k=0}^{\infty} |z_n| = \sum_{k=0}^{\infty} \sqrt{x_n^2 + y_n^2} < \infty\]

Q: Power Series
A: Series of the form         \[\sum_{k=0}^{\infty} a_k x^k \quad a_k, x \in \mathbb R\]

Q: Radius of Convergence
A: Let \(\sum_{k=0}^{\infty} a_k x^k\) be a power series. Let\[\rho = \limsup_{n\to \infty} \sqrt[n]{|a_n|} \text{ and } R = \frac{1}{\rho} \in [0,\infty]\]Where we used the convention that \(\frac{1}{0} = \infty\) and \(\frac{1}{\infty} = 0\). We call \(R\) the radius of convergence

Q: Exponential Function through series
A: Given \(x\in \mathbb R\), we define
        \[\exp(x) = \sum_{k=0}^{\infty} \frac{x^k}{k!}\]

Q: Odd function
A: \[f(x)=-f(-x)\]

Q: Even Function
A: \[f(x)=f(-x)\]

Q: Derivative
A: \[f'(x)=\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}\]

Q: Diffrentiation: Sum Rule
A: Let \(f,g\) be differentiable at \(x_0\in D\). Then\[(f+g)'(x_0)=f'(x_0)+g'(x_0)\]

Q: Diffrentiation: Product Rule
A: Let \(f,g\) be differentiable at \(x_0\in D\). Then\[(f\cdot g)'(x_0)=f'(x_0)\cdot g(x_0)+f(x_0)\cdot g'(x_0)\]

Q: Diffrentiation: Chain Rule
A: Given \(f: D\to E, g:E\to\mathbb R\), diffrentiable at \(x_0, f(x_0)\) then\[(g\circ f)'(x_0)=g'(f(x_0))\cdot f'(x_0)\]

Q: Quotient Rule
A: \[\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}\]

Q: Inverse Function Theorem (Derivatives)
A: \[\left(f^{-1}\right)'(y_0)=\frac{1}{f'\circ f^{-1}(y_0)}\]

Q: Local Maximum
A: \(x_0\in D\) is a local maximum if \(f(x_0)\ge f(x) \forall x\in(x_0-\delta,x_0+\delta)\)

Q: Rolle's Theorem
A: Given \(f: [a,b]\to \mathbb R\) continuous and diffrentiable on \((a,b)\) with \(f(a)=f(b)\), then \(\exists \xi\in(a,b)\) such that \(f'(\xi)=0\)

Q: Mean Value Theorem
A: Given \(f: [a,b]\to \mathbb R\) continuous and diffrentiable on \((a,b)\), then \(\exists \xi\in(a,b)\) such that \[f'(\xi)=\frac{f(b)-f(a)}{b-a}\]

Q: Convexity (t-Variant)
A: \(f\) is convex if \(\forall a,b\in I, \forall t\in (0,1)\)\[f(ta+(1-t)b)\le t\cdot f(a)+(1-t)f(b)\]

Q: Convexity (Cool variant)
A: \(f:I\to \mathbb R\) is convex if \(\forall a,b\in I\)\[\frac{f(b)-f(x)}{b-x}\ge \frac{f(x)-f(a)}{x-a} \quad\forall x\in(a,b)\]

Q: Partition
A: A collection \(\mathcal P\) of subsets of \(X\) such that\[X=\bigcup_{A\in\mathcal P}A \text{ and } A\cap B=\emptyset \forall A,B\in \mathcal P\]

Q: Decomposition
A: A family of points in an interval \([a,b]\) such that\[a=x_0<x_1<\cdots < x_n=b\]

Q: Step Function
A: A function \(f: [a,b]\to\mathbb R\) such that \(\exists\) a decomposition \(\{x_0,\cdots x_n\}\) of \([a,b]\) such that \(f|_{(x_k,x_{k+1})}\) is constant \(\forall k\).

Q: Integral of a step function
A: Given step function \(f\) with decomposition \(\{a=x_0<\cdots<x_n=b\}\) has integral\[\int\limits_a^b f(x)\mathrm dx\coloneqq\sum_{k=1}^nc_k(x_k-x_{k-1})\]

Q: Riemann Integrability
A: Given \(\mathcal L(f) \) as the set of lower step functions and \(\mathcal U(f)\) as the set of upper functions, then
\(f\) is Riemann integrable if \[\sup\mathcal L(f) = \inf \mathcal U(f) =\int\limits_a^b f(x)\mathrm dx\]

Q: Modulus and the Integral (Triangle Inequality Flavour)
A: \[\left|\int\limits_a^b f(x)\mathrm dx\right|\le\int\limits_a^b|f(x)|\mathrm dx\]

Q: Which two sets of functions are Riemann-Integrable
A: - Monotone Functions
- Continuous Functions

Q: Primitive
A: Given \(I\subseteq \mathbb R\) interval, \(f: I \to \mathbb R\). We say that \(F: I \to \mathbb R\) is a primitive of \(f\) if\[F'(x) = f(x) \quad \forall x\in I\]

Q: Fundamental Theorem of Calculus
A: Let \(f: [a,b] \to \mathbb R\) be continuous. Then, \(\forall C\in \mathbb R\), the function\[F(x) = \int_a^x f(t) \mathrm d t + C\]is a primitive of \(f\). Also, all primitives are of this form.

Q: Integral Test for Series
A: Let \(f:[0,\infty) \to \mathrm R_{\ge 0}\) be decreasing. Then,
\[\sum_{k=1}^{n+1} f(k) \le \int_0^{n+1} f(x) \mathrm d x \le \sum_{k=0}^n f(k).\]
In particular,
\[\sum_{k=1}^{\infty} f(k) \le \int_0^{\infty} f(x) \mathrm d x \le \sum_{k=0}^\infty f(k).\]

Q: Integration under Convergence
A: Given \(f_n:[a,b]\to\mathbb R\) integrable function converging uniformly to \(f\), then \(f\) is integrable and\[\int_a^bf_n(x)\mathrm dx=\int_a^bf(x)\mathrm dx\]

Q: Gamma Function
A: Given \(s>0\), define\[\Gamma(s)=\int_0^\infty x^{s-1}e^{-x}\mathrm dx\]

Q: Taylor Approximation
A: Given \(f: I \to \mathbb R\) \(n\) times differentiable, \(x_0\in I\) we define the \(n\)-th Taylor approximation of \(f\) at \(x_0\) as
\[P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(x_0)}{k!} (x - x_0)^k.\]

Q: Integral Remainder
A: Let \(n\ge 1, f:[a,b]\to \mathbb R\) be \(n\) times continously differentiable, \(x_0\in [a,b]\). Then
\[f(x) = P_{n-1}(x) + \int_{x_0}^x f^{(n)}(t)\cdot \frac{(x-t)^{n-1}}{(n-1)!} \mathrm d t.\]

Q: Lagrange Remainder
A: Let \(n\ge 1, f:[a,b]\to \mathbb R\) \(n\)-times differentiable, \(x_0\in [a,b]\). Then \(\forall x\in [a,b]\,\exists \xi_L\in(x_0,x)\) such that
\[f(x) = P_{n-1}(x) + \frac{f^{(n)}(\xi_L)}{n!} (x - x_0)^n.\]