True or False Every power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}(x-c) ^{k}$ converges for at least one number.

True or False Let $b_{n}$ denote the $n$th term of the power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$. If $\lim\limits_{n\,\rightarrow \,\infty }\,\left\vert \dfrac{b_{n+1}}{b_{n}} \right\vert \lt 1$ for every number $x$, then $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ is absolutely convergent on the interval $( -\infty ,\,\infty )$.

True or False If the radius of convergence of a power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ is $0$, then the power series converges only for $x=0$.

True or False If a power series converges at one endpoint of its interval of convergence, then it must converge at its other endpoint.

True or False The power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ and $\sum\limits_{k\,=\,0}^{\infty }a_{k}(x-3)^{k}$ have the same radius of convergence.

True or False The power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ and $\sum\limits_{k\,=\,0}^{\infty }a_{k}(x-3)^{k}$ have the same interval of convergence.

True or False If the power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ converges for $x=8$, then it converges for $x=-8$.

True or False If the power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ converges for $x=3$, then it converges for $x=1$.

True or False If the power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ converges for $x=-4$, then it converges for $x=3$.

True or False If the power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ converges for $x=3$, then it diverges for $x=5$.

True or False A possible interval of convergence for the power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ is $[-2,4]$.

True or False If the power series $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ diverges for a number $x_{1}$, then it converges for all numbers $x$ for which $\vert x\vert <\vert x_{1}\vert$.

$\sum\limits_{k\,=\,0}^{\infty} kx^{k}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{kx^{k}}{3^{k}}$

$\sum\limits_{k\,=\,0}^{\infty }\dfrac{(x+1)^{k}}{3^{k}}$

$\sum\limits_{k=1}^{\infty}\dfrac{(x-2) ^{k}}{k^{2}}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{2^{k}(k+1) }$

$\sum\limits_{k\,=\,0}^{\infty}(-1) ^{k}\dfrac{x^{k}}{2^{k}(k+1) }$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{k+5}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{1+k^{2}}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{k^{2}x^{k}}{3^{k}}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{2^{k}x^{k}}{3^{k}}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{kx^{k}}{2k+1}$

$\sum\limits_{k\,=\,0}^{\infty}(6x)^{k}$

$\sum\limits_{k\,=\,0}^{\infty}(x-3)^{k}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{k(2x)^{k}}{3^{k}}$

$\sum\limits_{k=1}^{\infty}\dfrac{x^{k}}{k^{3}}$

$\sum\limits_{k=2}^{\infty}\dfrac{x^{k}}{\ln k}$

$\sum\limits_{k=1}^{\infty}\dfrac{(x-2)^{k}}{k^{3}}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{k(x-2) ^{k}}{3^{k}}$

$\!\sum\limits_{k\,=\,0}^{\infty}\dfrac{(-1)^{k}}{(2k+1)!}x^{2k+1}$

$\sum\limits_{k=1}^{\infty}(kx)^{k}$

$\sum\limits_{k=1}^{\infty}\dfrac{kx^{k}}{\ln (k+1)}$

$\sum\limits_{k=1}^{\infty}\dfrac{x^{k}}{\ln (k+1)}$

$\sum\limits_{k=0}^{\infty}\dfrac{k(k+1)x^{k}}{4^{k}}$

$\sum\limits_{k=1}^{\infty}\dfrac{(-1)^{k}(x-5)^{k}}{k(k+1)}$

$\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{(x-3)^{2k}}{9^{k}}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{e^{k}}$

$\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{(2x)^{k}}{k!}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{(x+1)^{k}}{k!}$

$\sum\limits_{k\,=\,0}^{\infty}(-1)^{k}\dfrac{(x-1)^{4k}}{k!}$

$\sum\limits_{k=1}^{\infty}\dfrac{(x+1)^{k}}{k(k+1)(k+2)}$

$\sum\limits_{k=1}^{\infty}\dfrac{k^{k}x^{k}}{k!}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{3^{k}(x-2)^{k}}{k!}$

A function $f$ is defined by the power series $f(x) =\sum\limits_{k\,=\,0}^{\infty }\dfrac{x^{k}}{3^{k}}$.

A function $f$ is defined by the power series $f(x)=\sum\limits_{k\,=\,0}^{\infty }(-1) ^{k}\left(\dfrac{x}{2}\right)^{k}$.

A function $f$ is defined by the power series $f(x)=\sum\limits_{k\,=\,0}^{\infty }\dfrac{(x-2) ^{k}}{2^{k}}$.

A function $f$ is defined by the power series $f(x) =\sum\limits_{k\,=\,0}^{\infty }(-1) ^{k}(x+3)^{k}$.

If $\sum\limits_{k=0}^{\infty }a_{k\,}x^{k}$ converges for $x=3$, what, if anything, can be said about the convergence at $x=2$? Can anything be said about the convergence at $x=5$?

If $\sum\limits_{k=0}^{\infty }a_{k}(x-2)^{k}$ converges for $x=6$, at what other numbers $x$ must the series necessarily converge?

If the series $\sum\limits_{k=0}^{\infty }\,a_{k}x^{k}$ converges for $x=6$ and diverges for $x=-8$, what, if anything, can be said about the truth of the following statements?

If the radius of convergence of the power series $\sum\limits_{k=0}^{\infty }\,a_{k}(x-3) ^{k}$ is $R=5$, what, if anything, can be said about the truth of the following statements?

$f(x) =\dfrac{1}{1+x^{3}}$

$f(x) =\dfrac{1}{1-x^{2}}$

$f(x) =\dfrac{1}{6-2x}$

$f(x) =\dfrac{4}{x+2}$

$f(x) =\dfrac{x}{1+x^{3}}$

$f(x) =\dfrac{4x^{2}}{x+2}$

$f(x) =\sum\limits_{k\,=\,0}^{\infty}\dfrac{(-1) ^{k}x^{2k+1}}{(2k+1) !}$

$f(x)=\sum\limits_{k\,=\,0}^{\infty}\dfrac{(-1)^{k}x^{2k}}{(2k) !}$

$f(x)=\sum\limits_{k\,=\,0}^{\infty}\dfrac{x^{k}}{k!}$

$f(x)=\sum\limits_{k\,=\,0}^{\infty}\dfrac{(-1)^{k}x^{k}}{k!}$

$f(x) =\dfrac{1}{(1+x)^{2}}$

$f(x) =\dfrac{1}{(1-x)^{3}}$

$f(x)=\dfrac{2}{3 (1-x) ^{2}}$

$f(x) =\dfrac{1}{(1-x) ^{4}}$

$f(x) =\ln \left( \dfrac{1}{1+x}\right)$

$f(x) =\ln (1-2x)$

$f(x) =\ln (1-x^{2})$

$f(x) =\ln (1+x^{2})$

$\sum\limits_{k=1}^{\infty}\dfrac{x^k}{k}$

$\sum\limits_{k=1}^{\infty}\dfrac{(x-4)^{k}}{k}$

$\sum\limits_{k=1}^{\infty}\dfrac{x^{k}}{2k+1}$

$\sum\limits_{k=1}^{\infty}\dfrac{x^{k}}{k^{2}}$

$\sum\limits_{k\,=\,0}^{\infty}x^{k^{2}}$

$\sum\limits_{k=1}^{\infty}\dfrac{k^{a}}{a^{k}}(x-a)^{k}, \quad a\neq 0$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{(k!)^{2}}{(2k)!}(x-1)^{k}$

$\sum\limits_{k\,=\,0}^{\infty}\dfrac{\sqrt{k!}}{(2k)!}x^{k}$

Use the power series found in Problem 80 to get an approximation for $\ln 2$ correct to three decimal places.

Use the first 1000 terms of Gregory’s series to approximate $\dfrac{\pi }{4}$. What is the approximation for $\pi ?$

If $R>0$ is the radius of convergence of $\sum\limits_{k=1}^{\infty }a_{k}x^{k}$, show that $\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert =\dfrac{1}{R}$, provided this limit exists.

If $R$ is the radius of convergence of $\sum\limits_{k=1}^{\infty }a_{k}x^{k}$, show that the radius of convergence of $\sum\limits_{k=1}^{\infty }a_{k}x^{2k}$ is $\sqrt{R}$.

Prove that if a power series is absolutely convergent at one endpoint of its interval of convergence, then the power series is absolutely convergent at the other endpoint.

Suppose $\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ converges for $\vert x\vert <R$ and that $\lim\limits_{n\rightarrow \infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert$ exists. Show that $\sum\limits_{k=1}^{\infty }ka_{k}x^{k-1}$ and $\sum\limits_{k\,=\,0}^{\infty } \dfrac{a_{k}}{k+1}x^{k+1}$ also converge for $\vert x\vert <R$.

Consider the differential equation $$(1+x^{2})\,y^{\prime \prime} -4xy^\prime +6y=0$$

Assuming there is a solution $y(x)=\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ , substitute and obtain a formula for $a_{k}$. Your answer should have the form $$y(x)=a_{0}(1-3x^{2}) +a_{1}\!\left( x-\dfrac{1}{3}x^{3}\right)\qquad a_{0},~a_{1} \hbox{ real numbers}$$

If the series $\sum\limits_{k=0}^{\infty }a_{k}3^{k}$ converges, show that the series $\sum\limits_{k=1}^{\infty }ka_{k}2^{k}$ also converges.

Find the interval of convergence of the series $\sum\limits_{k=1}^{\infty }\dfrac{(x-2)^{k}}{k(3^{k})}$.

Let a power series $S(x)$ be convergent for $\vert x\vert <R$. Assume that $S(x)=\sum\limits_{k\,=\,0}^{\infty }a_{k}x^{k}$ with partial sums $S_{n}(x)=\sum\limits_{k\,=\,0}^{n}a_{k}x^{k}$. Suppose for any number $\varepsilon >0$, there is a number $N$ so that when $n>N$, $\vert S(x)-S_{n}(x) \vert <\dfrac{\varepsilon }{3}$ for all $\vert x\vert <R$. Show that $S(x)$ is continuous for all $\vert x\vert <R$.

Find the power series in $x$, denoted by $f(x)$, for which $f^{\prime \prime} (x) + f(x) =0$ and $f(0) =0$, $f^\prime (0) =1$. What is the radius of convergence of the series?

The Bessel function of order $m$ of the first kind, where $m$ is a nonnegative integer, is defined as $$J_{m}(x) =\sum\limits_{k\,=\,0}^{\infty }(-1) ^{k} \dfrac{1}{(k+m) !\, k!}\left( \dfrac{x}{2}\right) ^{2k+m}$$

Show that: