Power Series

• A power series (centered at $c$) is a series of the form $\sum _{n=0}^{\infty }{a}_n(x-c{)}^n$ where $(a_n)$ is a sequence of real numbers and $c$ is a real number.
  • The number $c$ is called the center of the power series.
  • The constants $a_0,a_1,a_2,\dots$ are called the coefficients of the power series.
  • A power series is actually a series of functions, where $f_n(x)=a_n(x-c{)}^n$.
  • A power series is a function of $x$, and it is defined for all $x$ in the interval of convergence.
  • In this discussion, we will talk about power series centered at $0$, of the form $\sum _{n=0}^{\infty }{a}_n{x}^n$. But the results can be generalized to power series centered at any real number $c$, by substituting $y=x-c$.

Radius of Convergence

Given a power series $\sum _{n=0}^{\infty }{a}_n{x}^n$

• (6.10) Let’s denote $C=\underset{n\to \infty }{\mathrm{limsup}}\sqrt[n]{|a_n|}$.
  • $R=\{\begin{array}{ll}0& \text{if }C=\infty \\ \infty & \text{if }C=0\\ \frac{1}{C}& \text{if }0<C<\infty \end{array}$ is called the radius of convergence of the power series.
  • (a.) The series converges absolutely for all real number $x$ in $(-R,R)$.
  • (b.) The series converges uniformly on every closed interval $[-r,r]$ where $0<r<R$.
• (6.11) $R=\underset{n\to \infty }{\lim }\left|\frac{a_n}{a_{n+1}}\right|$ is the radius of convergence. (given the limit exists, finite or infinite)
• (6.12) Let $R>0$ (or $R=\infty$) be the radius of convergence. And let $f(x)={\sum }_{n=0}^{\infty }{a}_n{x}^n$ be the function defined on $(-R,R)$. Then:
  • (a.) The function $f$ is continuous on $(-R,R)$. (if the series converges at $R$ or $-R$, then $f$ is continuous at $R$ or $-R$, respectively.)
  • (b.) for all $0<r<R$, the function $f$ is integrable on $[-r,r]$. And for all $x\in (-R,R)$, we have ${\int }_0^{x}f(t)dt={\sum }_{n=0}^{\infty }\frac{a_n}{n+1}{x}^{n+1}$.
  • (c.) The function $f$ is continuously differentiable on $(-R,R)$, and for all $x\in (-R,R)$, we have $f^{\prime }(x)={\sum }_{n=0}^{\infty }{a}_n(n+1)x^n={\sum }_{n=1}^{\infty }n{a}_n{x}^{n-1}$.
  • The power series in (b.) and (c.) have the same radius of convergence $R$ as the original power series.
• (6.13, Abel’s Theorem) Let $0<R\in \mathbb{R}$ be the radius of convergence, and let $f(x)={\sum }_{n=0}^{\infty }{a}_n{x}^n$ be the function defined on $(-R,R)$. If the series converges at $x=R$ (or $x=-R$), then the function $f$ is continuous at $x=R$ from the left, i.e. $\underset{x\to R^{-}}{\lim }f(x)=f(R)$ (or $\underset{x\to -R^{+}}{\lim }f(x)=f(-R)$, respectively).
• (p210) A power series with finite radius of convergence $R>0$ uniformly converges on the domain of convergence if and only if the series converges at the endpoints $x=R$ and $x=-R$.
• (6.14a) $\forall x\in (-r,r),f(x)=\sum _{n=0}^{\infty }{a}_n{x}^n⟹f$ is infinitely differentiable on $(-r,r)$
• (6.14b) $\forall x\in (-r,r),f(x)=\sum _{n=0}^{\infty }{a}_n(x-c{)}^n⟹\forall n\ge 0,a_n=\frac{f^{(n)}(c)}{n!}$
  • $\forall x\in (-r,r),f(x)=\sum _{n=0}^{\infty }{a}_n{x}^n⟹\forall n\ge 0,a_n=\frac{f^{(n)}(0)}{n!}$

Some Theorems

not-in-coursetodo

• If the power series $\sum _{n=0}^{\infty }{a}_n{x}^n$ converges in $x_0$, then:
  • the series converges absolutely for all $x$ such that $|x|<|x_0|$.
  • The series converges uniformly on every closed interval $[-r,r]$ where $0<r<|x_0|$.
  • (Cauchy-Hadamard) $R=\frac{1}{\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}}$ is the radius of convergence. given the limit exists.
  • (Cauchy-Hadamard, Stonger Version) $R=\frac{1}{\underset{n\to \infty }{\mathrm{limsup}}\sqrt[n]{|a_n|}}$ is the radius of convergence. (the limit always exists)
  • If the limit is $\infty$, then $R=0$, and if the limit is $0$, then $R=\infty$.
  • The series converges in $R$ (or $-R$) if and only if the series converges uniformly on $[0,R]$ (or $[-R,0]$, respectively).
• Given a power series $f(x)={\sum }_{n=0}^{\infty }{a}_n{x}^n$ on $[-R,R]$ that converges to $f$ on $[-R,R]$. Then:
  • For all $x\in (-R,R)$, we have $f^{\prime }(x)={\sum }_{n=1}^{\infty }{a}_{n}n(x{)}^{n-1}$. And radius of convergence of $f^{\prime }$ is $R$ as well. (it’s possible that the series of $f$ converges in $\pm R$, but the series of $f^{\prime }$ converges does not.)
  • For all $x\in (-R,R)$, we have ${\int }_0^{x}f(t)dt={\sum }_{n=0}^{\infty }\frac{a_n}{n+1}{x}^{n+1}$. And radius of convergence of ${\int }_0^{x}f(t)dt$ is $R$ as well. (it’s possible that the series of $f$ converges in $\pm R$, but the series of ${\int }_0^{x}f(t)dt$ converges does not.)

Operations

• Addition and Subtraction: Given two power series $f(x)=\sum a_n(x-c{)}^n$ and $g(x)=\sum b_n(x-c{)}^n$, then $f(x)\pm g(x)=\sum (a_n\pm b_n)(x-c{)}^n$.

  • The radius of convergence of $f(x)\pm g(x)$ is at least the minimum of the radii of convergence of $f(x)$ and $g(x)$.

• Multiplication and Division:todo

• Differentiation and Integration:

  • $f^{\prime }(x)={\sum }_{n=1}^{\infty }n{a}_n(x-c{)}^{n-1}={\sum }_{n=0}^{\infty }{a}_{n+1}(n+1)(x-c{)}^n$.
  • $\int f(x)dx={\sum }_{n=0}^{\infty }\frac{a_n}{n+1}(x-c{)}^{n+1}$.
  • The radius of convergence of $f^{\prime }(x)$ and $\int f(x)dx$ is the same as the radius of convergence of $f(x)$.

• todo Given power series $f(x)={\sum }_{n=0}^{\infty }{a}_n{x}^n$ and $g(x)={\sum }_{n=0}^{\infty }{b}_n(x{)}^n$ that converges to $f$ and $g$, respectively. Then:

  • (a.) $\sum c{a}_n{x}^n$ converges to $cf(x)$.
  • (b.) $\sum (a_n+b_n)x^n$ converges to $f(x)+g(x)$.
  • (c.) $a_n={\sum }_{k=0}^n{a}_k{b}_{n-k}$, then $\sum a_n{x}^n$ converges to $f(x)g(x)$. (Cauchy product)
  • (d.)todo composition of power series

Taylor Series

• The Taylor series of an infinitely differentiable function $f$ at a point $a$, is the power series $\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$.
  • When $a=0$ the series is called the Maclaurin series of $f$
  • The partial sum formed by the first $n+1$ terms of a Taylor series is a polynomial of degree $n$ that is called the $n$th Taylor Polynomial of $f$ at $a$. (or the $n$th Maclaurin polynomial of $f$)

Representing Functions

• (6.15) Given $f(x)$ is infinitely differentiable on $(-r,r)$, then the following are equivalent:

  • The function $f$ can be represented by a power series on $(-r,r)$
  • There exists a sequence $(a_n)$ such that for all $x\in (-r,r)$, we have $f(x)=\sum _{n=0}^{\infty }{a}_n{x}^n$
  • For all $x\in (-r,r)$, the Taylor series of $f$ at $0$ converges to $f$ at $x$.
  • For all $x\in (-r,r)$, we have $\underset{n\to \infty }{\lim }{S}_n(x)=f(x)$ (where $S_n(x)={\sum }_{k=0}^n{a}_k{x}^k$ is the $n$-th partial sum of the power series of $f$ at $x$).
  • For all $x\in (-r,r)$, we have $f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n$
  • For all $x\in (-r,r)$, we have $\underset{n\to \infty }{\lim }{R}_n(x)=0$ (where $R_n(x)=\frac{f^{(n+1)}(x)}{(n+1)!}$ is the remainder of the $n$-th Taylor polynomial of $f$ at $x$.)
  • ניתנת לפיתוח לטור חזקות

• If the function $f$ can be represented by a power series on $(-r,r)$, then:

  • $f$ is infinitely differentiable on $(-r,r)$.
  • The power series is unique, and the coefficients given by $a_n=\frac{f^{(n)}(0)}{n!}$.

• not-in-course Let $I\subset \mathbb{R}$ be an open interval. A function $f:I\to \mathbb{R}$ is (real) analytic if for all $a\in I$, there exists some $r>0$ and a sequence of coefficients $(a_n)$ such that $(a-r,a+r)\subset I$ and $f(x)={\sum }_{n=0}^{\infty }{a}_n(x-a{)}^n$. (todo this is stronger than being infinitely differentiable, and weaker than being represented by one power series on the whole interval. i.e. represented by a power series $⟹$ analytic $⟹$ infinitely differentiable)

• todo check ifnot-in-course - If $f$ infinitely differentiable on $(-R,R)$, and there exists $M$ such that $|f^{(n)}(x)|\le M$ for all $x\in (-R,R)$ and $n\in \mathbb{N}$, then $f$ can be represented by a power series on $[-R,R]$. (todo check also if relate to taylor’s ineq.)

Taylor’s Theorem

• (wiki) Let $n\ge 1$ be an integer, and let $f:\mathbb{R}\to \mathbb{R}$ be a function that is $n$ times differentiable at the point $a\in \mathbb{R}$. Then there exists a function $R_n:\mathbb{R}\to \mathbb{R}$ such that for all $x\in \mathbb{R}$, we have:

  • $f(x)={\sum }_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k+R_n(x)$
  • $\underset{x\to a}{\lim }{R}_n(x)=0$

• Let $f$ be a function that is $n+1$ times on an interval $I$ containing $a$. Let $P_n(x)={\sum }_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k$ be the $n$th Taylor polynomial of $f$ at $a$. Then for all $x\in I$, there exists $c$ between $a$ and $x$ such that $R_n(x)=f(x)-P_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a{)}^{n+1}$.

  • Conclusion: If there exists $M$ such that $|f^{(n+1)}(x)|\le M$ for all $x\in I$, then $|R_n(x)|\le \frac{M}{(n+1)!}|x-a{|}^{n+1}$.

Estimates for the remainder

todo it’s from wikipedia

Suppose that $f$ is $n+1$ times continuously differentiable in an interval $I$ containing $a$.

1. If there exists $q,Q$ such that $q\le f^{(n+1)}(x)\le Q$ for all $x\in I$, then for all $x\in I$, we have:
   • $q\frac{(x-a{)}^{n+1}}{(n+1)!}\le R_n(x)\le Q\frac{(x-a{)}^{n+1}}{(n+1)!}$ if $x>a$.
   • $Q\frac{(x-a{)}^{n+1}}{(n+1)!}\le R_n(x)\le q\frac{(x-a{)}^{n+1}}{(n+1)!}$ if $x<a$.
2. If $f^{(n+1)}\le M$ for all $x\in I=(a-r,a+r)$, (with some $r>0$), then for all $x\in I$, we have $|R_n(x)|\le M\frac{|x-a{|}^{n+1}}{(n+1)!}\le M\frac{r^{n+1}}{(n+1)!}$. (Taylor’s Inequality) (see ‘uniform convergence’)

Taylor Polynomial

• Let $f$ be a n-times differentiable function,

  • The polynomial $P_n(x)=\sum _{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k$ is called the $n$th order Taylor polynomial of $f$ centered at $a$.
    • When $a=0$, the polynomial $P_n(x)={\sum }_{k=0}^n\frac{f^{(k)}(0)}{k!}{x}^k$ is called also the Maclaurin polynomial of $f$
    • Taylor Polynomial is partial sum of Taylor Series
  • The function $R_n(x)=f(x)-P_n(x)$ is called the $n$-th remainder of $f$ at $a$
  • The representation $f(x)=P_n(x)+R_n(x)$ of $f$ is called $n$th order Taylor’s formula with remainder of $f$ centered at $a$

• $P_n^{(m)}(a)=f^{(m)}(a)$ (for all $m=0,\dots ,n$)

• (q4.11) Let $f(x)=x^k$ for $k\in \mathbb{N}$, and $P_n(x)$ Maclaurin polynomial of $f$, and $R_n(x)$ the reminder, then

  • for all $k>n$, $f(x)=R_n(x)$
  • for all $k\le n$, $f(x)=P_n(x)$

• (4.7) If $f$ is a $n$ times differentiable (at $a$) function where $R_n(x)$ is its n-th remainder at $a$, then $\underset{x\to a}{\lim }\frac{R_n(x)}{(x-a{)}^n}=0$

• (4.8) (Uniqueness of Taylor Polynomial) If $f$ is a $n$ times differentiable (at $a$) function and $P_n$ is a polynomial and $\underset{x\to a}{\lim }\frac{f(x)-P_n(x)}{(x-a)}=\underset{x\to a}{\lim }\frac{R_n(x)}{(x-a)}=0$ then $P_n$ is the $n$th order Taylor polynomial of $f$ centered at $a$

Remainder Term

Lagrange’s Form

• (4.4) if $f$ is a $n+1$ times differentiable function on $I$ where $a\in I$ and $R_n(x)$ is the n-th remainder of $f$ at $a$. then:

  • (Lagrange’s Form) there exists a number $c$ between $a$ and $x$ s.t: $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a{)}^{n+1}$

• (page-70) if $f$ is continuous on $[a,b]$ (or $[b,a]$), and $n+1$ times differentiable in $(a,b)$ (or $(b,a)$), and and $n$ times differentiable in $a$ such that $f^{(n)}(x)$ is continuous at $a$, then there exists $c$ between $a$ and $b$ for which $R_n(b)=\frac{f^{(n+1)}(c)}{(n+1)!}(b-a{)}^{n+1}$ that is $f(b)=\sum _{k=0}^n\frac{f^{(n)}(a)}{k!}(b-a{)}^k+\frac{f^{(n+1)}(c)}{(n+1)!}(b-a{)}^{n+1}$

  • for $n=0$ it is Lagrange’s MVT

Cauchy’s Form

• (4.5) if $f$ is a $n+1$ times differentiable function on $I$ where $a\in I$ and $R_n(x)$ is the n-th remainder of $f$ at $a$, then for every $x\in I$ there exists a number $c$ between $a$ and $x$ s.t: $R_n(x)=(x-a)\cdot \frac{f^{(n+1)}(c)}{n!}(x-c{)}^n$

General Form

not-in-course

• (4.6) if $f$ is a $n+1$ times differentiable function on $I$ where $a\in I$ and $R_n(x)$ is the n-th remainder of $f$ at $a$. Let be $b\in I$ and $b\ne a$, and be a continuous function $\psi :[a,b]\to \mathbb{R}$, where $\psi$ is differentiable on $(a,b)$ and $\psi (a)=0$ and ${\psi }^{\prime }(t)\ne 0$ for all between $a$ and $b$. Then there exists a number $c$ between $a$ and $b$ s.t: $R_n(b)=\frac{\psi (b)}{{\psi }^{\prime }(c)}\cdot \frac{f^{(n+1)}(c)}{n!}(b-c{)}^n$.
• Schlömilch’s Form (see q4.37)

Exmaples

• $\sin x=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{(2n+1)!}{x}^{2n+1}$
• $\cos x=\sum _{n=0}^{\infty }\frac{(-1{)}^n}{(2n)!}{x}^{2n}$
• $e^x=\sum _{n=0}^{\infty }\frac{1}{n!}{x}^n$
• $e^{-x}\sum _{n=0}^{\infty }\frac{(-1{)}^n}{n!}{x}^n$
• $\ln (1+x)=\sum _{n=1}^{\infty }\frac{(-1{)}^{n+1}}{n}{x}^n$
• $\ln (1-x)=-\sum _{n=1}^{\infty }\frac{1}{n}{x}^n$