Sequences and series - convergence tests, power series, and Taylor expansions.
These practice problems cover infinite sequences and series: whether a series converges, what it sums to when a closed form exists, and how well a partial sum approximates the total. The set belongs after a calculus course that already includes derivatives. Power series, Taylor expansions, uniform convergence, and Fourier series are all in the list. Every solution is written step by step.
Geometric and Telescoping Series is the chapter to open first, because those series can be summed exactly. Convergence Tests covers the integral, comparison, ratio, root, and alternating tests, and the decision of which test fits. Numerical and Functional Series and Uniform Convergence move from series of numbers to series of functions. Remainders and Fourier Series includes Taylor remainders and Fourier expansions on an interval. Infinite Products is a short, harder coda.
If you only need the Taylor polynomial of a function you can differentiate, the Taylor Series chapter in Differential Calculus is the lighter set. Sequence limits that fail here are practiced in Limits. Fourier series built while separating variables also appear in Partial Differential Equations.
Problem 1
HardGeometric and Telescoping Series
Evaluate the partial sum $S_n$ and the sum $S = \lim_{n \to \infty} S_n$ for the series:
$$
\sum_{n=1}^{\infty} q^n.
$$
Evaluate the sum $S$ for the series:
$$
S = \sum_{n=2}^{\infty} \frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}},
$$
where the positive numbers $a_n$ form an arithmetic progression.
Evaluate the sums for the following series, for $|q|<1$:
$$
1^{\circ}\; S_1 = \sum_{n=1}^{\infty} q^n \sin(nx); \qquad 2^{\circ}\; S_2 = \sum_{n=1}^{\infty} q^n \cos(nx).
$$
The regions bounded by the curve $y = e^{-x}\sqrt{\sin x}$ $(x\ge0)$ and the $x$-axis are rotated about the $x$-axis. Find the sum of the volumes of the solids generated by this rotation.
Based on the general term $a_n$, conclude that the following series diverges:
$$
\sum_{n=0}^{\infty} \cos(n\alpha), \qquad -\frac{\pi}{2} < \alpha < \frac{\pi}{2}.
$$
Determine the limit to which the following expression tends as $n \to \infty$, for $\alpha > -1$:
$$
a_n = \frac{1}{n^{1+\alpha}}\sum_{k=1}^{n} k^\alpha.
$$
Using the Cauchy criterion, prove that the convergence of the series $\sum_{n=1}^{\infty} a_n$ implies the convergence of the series $\sum_{n=1}^{\infty} A_n$, where
$$
A_n = \sum_{i=p_n}^{p_{n+1}-1} a_i \qquad (p_1=1,\; p_n < p_{n+1},\; n=1,2,\ldots).
$$
Show that, in the setting of the previous problem:
$$
\sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} A_n,
$$
where $A_n = \sum_{i=p_n}^{p_{n+1}-1} a_i$ with $p_1=1$, $p_n
View solution
Problem 23
HardGeometric and Telescoping Series
Using the Cauchy criterion, prove that the following series converges:
$$
a_0 + \sum_{i=1}^{\infty} \frac{a_i}{b^i} \qquad (|a_i| < b,\; b>1).
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{1}{2n-1}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{1}{(2n-1)^2}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{1}{1+a^n} \qquad (a>0).
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{1}{n\sqrt{n+1}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{1}{\sqrt{3n(2n-1)}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt[3]{n}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{1}{(an+b)^p}, \qquad a>0.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} (\sqrt[n]{a}-1), \qquad a>1.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \left(a^{1/n} + a^{-1/n} - 2\right), \qquad a>0.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=2}^{\infty} \frac{1}{\sqrt{n}} \ln\left(\frac{n+1}{n-1}\right).
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \left(\frac{1}{n} - \ln\frac{n+1}{n}\right).
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \sin\frac{x}{n}, \qquad x\neq0.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \frac{n\sin\frac{1}{n}}{\sqrt{n^2+1}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \left(e^{\frac{1}{n+1}} - 1\right)^p.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=0}^{\infty} 2^n \sin\frac{1}{3^n}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=2}^{\infty} (\sqrt{n+1}-\sqrt{n})^a \ln\frac{n-1}{n+1}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=2}^{\infty} \frac{1}{(\ln n)^\alpha}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=3}^{\infty} \frac{1}{(\ln\ln n)^{\ln n}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=3}^{\infty} \ln\frac{\cosh\frac{\pi}{n}}{\cos\frac{\pi}{n}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=3}^{\infty} \frac{1}{(\ln n)^{\ln n}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=3}^{\infty} \frac{1}{(\ln n)^{\ln \ln n}}.
$$
Knowing that $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}$ converges for $\alpha>1$ and diverges for $\alpha\le1$, determine the convergence of:
$$
\sum_{n=1}^{\infty} \left[e^3 - \left(1+\frac{1}{n^2}\right)^{3n^2}\right]^p.
$$
If the series $\sum_{n=1}^{\infty} a_n$ with $a_n>0$ converges, prove that $\sum_{n=1}^{\infty} a_n^2$ also converges. Show by example that the converse is not true.
Given that $\sum_{n=1}^{\infty} a_n^2$ and $\sum_{n=1}^{\infty} b_n^2$ converge, prove that the following series converge:
$$
1^\circ\; \sum_{n=1}^{\infty}|a_nb_n|; \qquad 2^\circ\; \sum_{n=1}^{\infty}(a_n\pm b_n)^2; \qquad 3^\circ\; \sum_{n=1}^{\infty}\frac{|a_n|}{n}.
$$
Using Cauchy's root test, examine the convergence of:
$$
\sum_{n=1}^{\infty}\left(\frac{x}{a_n}\right)^n,
$$
where $\lim_{n\to\infty} a_n = a$, $x>0$, and $a_n>0$.
If D'Alembert's test (ratio test) resolves the question of convergence of a series, then Cauchy's test (root test) also resolves it. The converse is not true. Prove this.
Apply D'Alembert's and Cauchy's tests to the series $\sum_{n=1}^{\infty} a_n$ where
$$
a_{2k-1} = \frac{2^{k-1}}{3^{k-1}}, \qquad a_{2k} = \frac{2^{k-1}}{3^k} \qquad (k=1,2,\ldots),
$$
and confirm the second part of the theorem in Problem 78.
If $\varlimsup_{n\to\infty} \sqrt[n]{a_n} = q$ where $a_n\ge0$, then for $q<1$ the series is convergent and for $q>1$ the series is divergent. Prove this.
Using Cauchy's integral test, examine the convergence of:
$$
1^\circ\; \sum_{n=2}^{\infty} \frac{1}{n \ln^3 n \ln^2(\ln n)}; \qquad
2^\circ\; \sum_{n=2}^{\infty} \frac{1}{n \ln n \ln(\ln n)}.
$$
Let $S=\sum_{n=1}^{\infty} \frac{1}{n^\alpha}$ where $\alpha>1$.
$1^\circ$ Show that if we take $S_n=\sum_{k=1}^n \frac{1}{k^\alpha}$ instead of $S$, the error is less than $\dfrac{1}{\alpha-1} \cdot \dfrac{1}{n^{\alpha-1}}$.
$2^\circ$ With $\Sigma=S_n+\displaystyle\int_n^{\infty} \frac{dx}{x^\alpha}$, show that $S$ lies between $\Sigma-\dfrac1{2n^\alpha}$ and $\Sigma$.
$3^\circ$ Show that $\Sigma-\dfrac1{2n^\alpha}+\dfrac{\alpha}{12(n+1)^{\alpha+1}}<S<\Sigma-\dfrac1{2n^\alpha}+\dfrac{\alpha}{12(n-1)^{\alpha+1}}$.
If $a_n>a_{n+1}>0$ for all sufficiently large $n$, prove that the series $\sum_{n=1}^{\infty} a_n$ and the series $\sum_{n=1}^{\infty} 2^n a_{2^n}$ converge or diverge simultaneously.
It can be proven that if $\sum_{n=1}^{\infty} a_n=A$, $\sum_{n=1}^{\infty} b_n=B$, $\sum_{n=1}^{\infty} c_n=C$ where $c_n=\sum_{v=1}^n a_v b_{n-v+1}$, then $AB=C$ (assuming both original series converge absolutely).
Can Leibniz's test be applied to the series
$$
\frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}-1}-\frac{1}{\sqrt{3}+1}+\cdots+\frac{1}{\sqrt{n}-1}-\frac{1}{\sqrt{n}+1}+\cdots\,?
$$
Does the given series converge?
Given that $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}=\ln 2$, find the sum of the series
$$
1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\cdots+\frac{1}{2k-1}-\frac{1}{4k-2}-\frac{1}{4k}+\cdots
$$
which is obtained from the given series by rearrangement of its terms.
Using Dirichlet's test, prove the convergence of:
$$
\sum_{n=1}^{\infty} \frac{(-1)^{[\sqrt{n}]}}{n},
$$
where $[\sqrt n]$ denotes the greatest integer (floor) function.
Test the absolute and conditional convergence of the series:
$$
\sum_{n=1}^{\infty} \ln\!\left(1+\frac{\sin\frac{n\pi}4}{n^\alpha}\right), \qquad \alpha>0.
$$
Test the absolute and conditional convergence of the series:
$$
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\left(\sqrt n+(-1)^{n-1}\right)^a}, \qquad a>0.
$$
Investigate whether the sum (or difference) of two divergent series can itself converge. That is: if $\sum a_n$ and $\sum b_n$ both diverge, can $\sum(a_n+b_n)$ or $\sum(a_n-b_n)$ converge?
Investigate whether the Cauchy product $\sum c_n$, with $c_n=\sum_{k=0}^n a_k b_{n-k}$, of two convergent series $\sum a_n$ and $\sum b_n$ must itself converge, and whether absolute convergence of the factors is necessary for this.
A sequence is defined by $a_1=1$ and $\dfrac{a_{n+1}}{a_n}=\dfrac34+\dfrac{(-1)^n}2$ for $n\ge1$. Test the absolute and conditional convergence of $\sum_{n=1}^{\infty}(-1)^n a_n$, and find its sum if it converges.
Find the Cauchy product of the series $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n}{\sqrt{n+1}}$ with itself, and show that the resulting series does not converge although the original series converges.
Using the Cauchy product, show that $\displaystyle\sum_{n=2}^{\infty} \frac{n(n-1)}{2^{n-2}}=16$, then find the sum of the series $\displaystyle\sum_{n=1}^{\infty} \frac{n^2}{2^n}$.
Let $a_{ij}=\dfrac{i-j}{2^{i+j-2}}\dfrac{(i+j-3)!}{(i-1)!(j-1)!}$ for $i,j>1$, with $a_{i1}=2^{-(i-1)}$ for $i>1$, $a_{1j}=-2^{-(j-1)}$ for $j>1$, and $a_{11}=0$. Show that:
$$
\sum_{j=1}^{\infty}\sum_{i=1}^{\infty}a_{ij}=1, \qquad \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}=-1.
$$
Let $r<p$. Prove that if the series $\displaystyle\sum_{n=1}^{\infty}\left(\sum_{v=1}^{n}|u_v|^r\right)^{1/r}$ converges, then the series $\displaystyle\sum_{n=1}^{\infty}\left(\sum_{v=1}^{n}|u_v|^p\right)^{1/p}$ also converges.
Let $r<p$. Prove that if the series $\displaystyle\sum_{m=1}^{\infty}\left(\sum_{n=1}^{\infty}|u(n,m)|^r\right)^{1/r}$ converges, then the series $\displaystyle\sum_{m=1}^{\infty}\left(\sum_{n=1}^{\infty}|u(n,m)|^p\right)^{1/p}$ also converges. Can the sums $\sum_{m=1}^{\infty}$ and $\sum_{n=1}^{\infty}$ be interchanged and the statement still hold?
Let $\sum_{p=1}^{\infty}x_p^2=1$, $\sum_{p=1}^{\infty}y_p^2=1$, and suppose $\sum_{p,q=1}^{\infty}a_{pq}x_py_q$ converges. Show that the following are equivalent: (1) $\displaystyle\sum_{q=1}^{\infty}\left(\sum_{p=1}^{\infty}a_{pq}x_p\right)y_q$ converges; (2) $\displaystyle\sum_{p=1}^{\infty}\left|\sum_{q=1}^{\infty}a_{pq}y_q\right|^2$ converges.
An infinite matrix $A=(a_{ij})$ maps $(x_n)$ to $(x'_n)$ via $x'_n=\sum_{j=1}^{\infty}a_{nj}x_j$. Given that $\lim_{n\to\infty}a_{nk}=0$ for each fixed $k$, show that $A$ transforms every bounded sequence into a null sequence if and only if $\displaystyle\lim_{n\to\infty}\sum_{k=1}^{\infty}|a_{nk}|=0$.
With $I_n=\frac1\pi\int_{-\pi}^{\pi}t\left[\sum_{r=1}^n(-1)^r(x_r\cos rt-y_r\sin rt)\right]^2dt$, and using $\int_{-\pi}^{\pi}t\sin mt\,dt=\frac{(-1)^{m+1}2\pi}m$, $\int_{-\pi}^{\pi}t\cos mt\,dt=0$, show that $I_n=2(S_n-T_n)$ where $S_n=\sum_{p,q=1}^n\frac{x_py_q}{p+q}$ and $T_n=\sum_{p,q=1}^n{}'\frac{x_py_q}{p-q}$ (excluding $p=q$). If $S=\sum_{p,q=1}^{\infty}\frac{x_py_q}{p+q}$, $T=\sum_{p,q=1}^{\infty}{}'\frac{x_py_q}{p-q}$, and $\sum x_p^2=\sum y_q^2=1$, prove $|S|\le\pi$, $|T|\le2\pi$.
For $u_n(x)=x^n$ ($n=1,2,\ldots$): (1) show the sequence converges on $[0,1]$ and find its limit; (2) show it does not converge uniformly on $[0,1]$; (3) does it converge uniformly on $[0,a]$ for $a<1$?
Using the Cauchy criterion, test the uniform convergence of the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{nx}{(1+x)(1+2x)\cdots(1+nx)}$ on $[1,\infty)$.
If Newton's series $\displaystyle\sum_{n=0}^{\infty}a_n\frac{(x-1)(x-2)\cdots(x-n)}{n!}$ converges for $x=x_0$ (where $x_0\ne0,1,2,\ldots$), prove that it converges for every $x>x_0$.
Prove that if the series $\displaystyle\sum_{n=1}^{\infty}(-1)^na_ne^{-\tau\sum_{k=1}^n\frac1k}$ converges, then the series $\displaystyle\sum_{n=0}^{\infty}a_n\frac{(x-1)(x-2)\cdots(x-n)}{n!}$ converges (with $\tau=x$).
Prove that Newton's series $\displaystyle\sum_{n=0}^{\infty}a_n\frac{(x-1)(x-2)\cdots(x-n)}{n!}$ and Dirichlet's series $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^na_n}{n^x}$ have the same region of ordinary convergence and of absolute convergence.
Using the Weierstrass test, show that the series $\displaystyle\sum_{n=1}^{\infty}\frac{\cos nx}{n^a}$ and $\displaystyle\sum_{n=1}^{\infty}\frac{\sin nx}{n^a}$ converge uniformly on $(-\infty,\infty)$ for $a>1$.
Let $\varphi:\mathbb R\to\mathbb R$ be a bounded function. Using the Weierstrass test, show that the series $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2+[\varphi(x)]^2}$ converges uniformly on $(-\infty,\infty)$.
Using the Weierstrass test, show that the series $\displaystyle\sum_{n=1}^{\infty}\frac1{n^x}$ converges uniformly on $[a,\infty)$ for any fixed $a>1$, but not uniformly on $(1,\infty)$ itself.
Using the Weierstrass test, show that the series $\displaystyle\sum_{n=2}^{\infty}\ln\!\left(1+\frac{x^2}{n\ln^2n}\right)$ converges uniformly on $[-a,a]$ for any fixed $a>0$.
If $\displaystyle\sum_{n=1}^{\infty}|u_n(x)|$ converges uniformly on $[a,b]$, prove that $\displaystyle\sum_{n=1}^{\infty}u_n(x)$ converges uniformly on $[a,b]$.
Test the conditional, absolute, and uniform convergence of $\displaystyle\sum_{n=1}^{\infty}\left(\frac{x}{x+1}\right)^{\alpha n}$, $\alpha\in\mathbb R$.
Using $\displaystyle\sum_{n=0}^{\infty}x^n=\frac1{1-x}$ ($|x|<1$), find the sums of $1^{\circ}\ 1+2x+3x^2+4x^3+\cdots$ and $2^{\circ}\ 1+3x+6x^2+10x^3+\cdots$
Determine the radius of convergence for $\displaystyle\sum_{n=1}^{\infty}\left(\frac{na^{n^2}}{n+2}\right)^p\frac{x^n}{n!}$, where $a>0$, $p\in\mathbb R$.
If $R_1,R_2$ are the radii of convergence of $\sum a_nx^n$ and $\sum b_nx^n$, what can be said about the radii of $1^{\circ}\ \sum(a_n+b_n)x^n$ and $2^{\circ}\ \sum a_nb_nx^n$?
Find the region of convergence for the hypergeometric series $1+\dfrac{\alpha\beta}{1\cdot\gamma}x+\dfrac{\alpha(\alpha+1)\beta(\beta+1)}{1\cdot2\cdot\gamma(\gamma+1)}x^2+\cdots$.
Expand $f(x)=\dfrac1{a-x}$ ($a\ne0$) in $1^{\circ}$ powers of $x$, $2^{\circ}$ powers of $(x-b)$ with $b\ne a$, $3^{\circ}$ powers of $1/x$. Determine the region of convergence in each case.
Using the power series expansions of $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^a$, expand $\ln(1+3x+2x^2)$ in a power series in $x$ and indicate when the expansion is valid.
Using the power series expansions of $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^a$, expand $\dfrac{x\cos\alpha-x^2}{1-2x\cos\alpha+x^2}$ in a power series in $x$ and indicate when the expansion is valid.
Using the power series expansions of $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^a$, expand $\dfrac{x\sin\alpha}{1-2x\cos\alpha+x^2}$ in a power series in $x$ and indicate when the expansion is valid.
Using the power series expansions of $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^a$, expand $\displaystyle\int_0^x\frac{\ln(1+t)}t\,dt$ in a power series in $x$ and indicate when the expansion is valid.
Using the power series expansions of $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^a$, expand $\displaystyle\int_1^x\frac{\arctan t}t\,dt$ in a power series in $x$ and indicate when the expansion is valid.
Using the power series expansions of $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $(1+x)^a$, expand $\displaystyle\int\frac{\cos x}x\,dx$ in a power series in $x$ and indicate when the expansion is valid.
By expanding the derivative $f'(x)$ in a power series in $x$ and integrating term by term, write the power series expansion in $x$ for $f(x)=\arctan x$.
By expanding the derivative $f'(x)$ in a power series in $x$ and integrating term by term, write the power series expansion in $x$ for $f(x)=\arcsin x$.
By expanding the derivative $f'(x)$ in a power series in $x$ and integrating term by term, write the power series expansion in $x$ for $f(x)=\ln\bigl(x+\sqrt{1+x^2}\bigr)$.
By expanding the derivative $f'(x)$ in a power series in $x$ and integrating term by term, write the power series expansion in $x$ for $f(x)=\arccos(1-2x^2)$.
Using appropriate power series, calculate to accuracy $0.0001$: $\cos10^\circ$; $\sin1^\circ$; $\sin\frac\pi4$; $\arcsin1$; $\arctan\frac15$; $\sqrt e$.
By expanding the integrand in a power series, calculate $\displaystyle\int_0^1\frac{\ln(1+x)}x\,dx$, given that $\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}}{n^2}=\dfrac{\pi^2}{12}$.
By expanding the integrand in a power series, calculate $\displaystyle\int_0^1\frac{\ln(1-x)}x\,dx$, given that $\sum_{n=1}^{\infty}\dfrac1{n^2}=\dfrac{\pi^2}6$.
Given $\int_0^{\pi/2}\sin^{2n}\varphi\,d\varphi=\dfrac{(2n-1)!!}{(2n)!!}\cdot\dfrac\pi2$, show the arc length of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}9=1$ equals $$ s = 10\pi\left\{1-\sum_{n=1}^{\infty}\left[\frac{(2n-1)!!}{(2n)!!}\right]^2\frac{(4/5)^{2n}}{2n-1}\right\}. $$
$1^\circ$ Show $I'(a)=\dfrac a2\displaystyle\sum_{k=1}^{\infty}\dfrac1{k^2+a^2}$. $2^\circ$ Using the partial-fraction expansion $\coth x=\dfrac1x+\displaystyle\sum_{n=1}^{\infty}\dfrac{2x}{x^2+n^2\pi^2}$, find $I(a)$ in elementary terms.
The coefficients $a_n$ of $\sum_{n=1}^{\infty}a_nx^n$ are the successive decimal digits of $\dfrac47$. Determine the radius of convergence and the sum.
$1^\circ$ Determine the interval of convergence of $f_\lambda(x)=\sum_{n=1}^{\infty}n^\lambda x^{n-1}$ and examine the endpoints. $2^\circ$ Show $f_{\lambda+1}(x)=\dfrac d{dx}[xf_\lambda(x)]$.
$1^\circ$ Find the radius of convergence depending on $a,\alpha$. $2^\circ$ For $\alpha=1$, $a\ge2$, examine the endpoints. $3^\circ$ For $\alpha=0$ and (briefly) $\alpha=-\tfrac13$, find the sum. $4^\circ$ Sum $\sum_{k=1}^{\infty}\dfrac{2^{2k}}{(2k)!}x^{2k}$.
$1^\circ$ Find the radius of convergence depending on $\alpha$. $2^\circ$ For $\alpha=-1$, examine the endpoints. $3^\circ$ For $\alpha=0$, find the sum.
$1^\circ$ Find the domain of convergence, for $x\ge0$, of $f(x)=\sum_{n=1}^{\infty}\dfrac{(-1)^{n-1}}{4n^2-1}\left(\dfrac{x-1}{x+1}\right)^{2n}$. $2^\circ$ Find a closed form for $f(x)$. $3^\circ$ Evaluate $\sum_{n=1}^{\infty}\dfrac{(-1)^n}{(4n^2-1)9^n}$.
Find the domain of convergence and the sums $f(r,\theta)=\sum_{n=1}^{\infty}r^n\cos(n\theta)$ and $\varphi(r,\theta)=\sum_{n=1}^{\infty}r^n\sin(n\theta)$.
Show that the function $f(x) = \dfrac{E(x)}{1+x^2}$ satisfies Dirichlet's conditions on every finite interval, where $E(x)=\lfloor x\rfloor$ denotes the integer part (floor function) of $x$.
$1^\circ$ Expand $f(x)=x$ in a Fourier series on $(-\pi,\pi)$. $2^\circ$ For which $x$ does the series equal $f(x)$? $3^\circ$ Express $\pi/2$ as a convergent numerical series. $4^\circ$ Find $\sum_{n=1}^\infty\frac{(-1)^{n-1}}{2n-1}$.
$1^\circ$ Expand the $2\pi$-periodic extension of $f(x)=x^2$ on $[-\pi,\pi]$ in a Fourier series. $2^\circ$ Find (a) $\sum_{n=1}^\infty\frac1{n^2}$ and (b) $\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}$.
$1^\circ$ Show that the Fourier coefficients of a $2l$-periodic function can be computed using $a_n=\frac1l\int_\lambda^{\lambda+2l}f(x)\cos\frac{n\pi x}l\,dx$, $b_n=\frac1l\int_\lambda^{\lambda+2l}f(x)\sin\frac{n\pi x}l\,dx$ for any $\lambda$. $2^\circ$ Expand $f(x)=\frac{\pi-x}2$ on $[0,2\pi]$, period $2\pi$, in a Fourier series.
$1^\circ$ Expand $f(x)=\arcsin(\sin x)$ in a Fourier series. $2^\circ$ Show that $$ \frac4\pi\sum_{n=1}^\infty\frac{(-1)^{n-1}}{(2n-1)^2}\sin((2n-1)x) = \sum_{n=1}^\infty\frac{(-1)^{n+1}}n\sin(2nx) = x $$ for $x\in\left(-\dfrac\pi2,\dfrac\pi2\right)$.
Expand $f(x)=e^{tx}$ on $(0,2\pi)$ (period $2\pi$) in a Fourier series, and show that $\pi\coth(\pi t)=\dfrac1t+2\sum_{n=1}^\infty\dfrac t{n^2+t^2}$, $t\ne0$.
By expanding in a power series in $\alpha$, find the Fourier series (in $x$) of $f(x)=\dfrac{\alpha\cos x-\alpha^2}{1-2\alpha\cos x+\alpha^2}$, $|\alpha|>1$.
Show that for $f(x)$ satisfying Dirichlet's conditions on $(-\pi,\pi)$ with continuous third derivative (and periodic boundary matching of $f,f',f''$), $$ a_n=\frac1{\pi n^3}\int_{-\pi}^\pi f'''(x)\sin(nx)\,dx, \qquad b_n=-\frac1{\pi n^3}\int_{-\pi}^\pi f'''(x)\cos(nx)\,dx. $$
$1^\circ$ Develop power series for $f(x)=e^{x\cot a}\cos x$, $\varphi(x)=e^{x\cot a}\sin x$. $2^\circ$ Sum $\sum_{k=1}^\infty\frac{\sin ka}{k!}$ and $\sum_{k=1}^\infty\frac{\cos kx}{k!}$. $3^\circ$ Develop Fourier series for $e^{\cos x}\cos(\sin x)$ and $e^{\cos x}\sin(\sin x)$.
Investigate the infinite product $\displaystyle\prod_{n=1}^{\infty}\left(1+\frac{(-1)^n}{a^n}\right)$ for $a>1$. (The source states a closed form of $a^{-\ln2}$; this closed form does not hold, as shown below — the correct value is derived instead.)
Investigate whether $\displaystyle\sum_{k=1}^{\infty}a_k=\prod_{k=1}^{\infty}(1-a_k)$ is possible for a sequence with $0
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Problem 384
MediumInfinite Products
Prove that the product $\displaystyle\prod_{n=1}^{\infty}(1\pm\alpha_n)$ (with $\alpha_n>0$) converges if and only if the series $\displaystyle\sum_{n=1}^{\infty}\alpha_n$ converges.
Show that Problem 384's criterion fails once the $\alpha_n$ do not have constant sign, by verifying that $$ \left(1+\frac1{\sqrt2}+\frac12\right)\left(1-\frac1{\sqrt2}\right)\left(1+\frac1{\sqrt3}+\frac13\right)\left(1-\frac1{\sqrt3}\right)\cdots $$ converges although the corresponding series $\left(\frac1{\sqrt2}+\frac12\right)+\left(-\frac1{\sqrt2}\right)+\left(\frac1{\sqrt3}+\frac13\right)+\left(-\frac1{\sqrt3}\right)+\cdots$ diverges.
Prove that from the convergence of $\displaystyle\prod_{n=1}^{\infty}p_n$ and $\displaystyle\prod_{n=1}^{\infty}q_n$ (to positive finite limits) it follows that $\displaystyle\prod_{n=1}^{\infty}p_nq_n$ and $\displaystyle\prod_{n=1}^{\infty}p_n/q_n$ also converge.
Show that $\displaystyle\Gamma(x)=\frac1x\prod_{n=1}^{\infty}\left(1+\frac1n\right)^x\left(1+\frac xn\right)^{-1}$ converges absolutely for every $x$ other than $0$ and the negative integers.
Prove that $\displaystyle\Gamma(x)=\lim_{n\to\infty}\frac{n!\,n^x}{x(x+1)(x+2)\cdots(x+n)}$. Then show $\Gamma(x+1)=x\Gamma(x)$, and verify $\Gamma(n+1)=n!$.
Starting from $\displaystyle\sin2x=2x\prod_{k=1}^{\infty}\left(1-\frac{4x^2}{k^2\pi^2}\right)$, show that $\displaystyle\cos x=\prod_{k=1}^{\infty}\left[1-\frac{4x^2}{(2k-1)^2\pi^2}\right]$.
Given that Euler's product $\sin x=x\prod_{k=1}^\infty\left(1-\dfrac{x^2}{k^2\pi^2}\right)$ holds for complex $x$, and using the relationship between $\sin x$ and $\sinh x$, show that $$ \sinh x = x\prod_{k=1}^{\infty}\left(1+\frac{x^2}{k^2\pi^2}\right). $$
Prove that $\displaystyle\frac1{\sin x}=\frac1x+2x\sum_{k=1}^{\infty}\frac{(-1)^k}{x^2-k^2\pi^2}$. From this, conclude that $\Gamma(x)\Gamma(1-x)=\dfrac\pi{\sin\pi x}$.