Numerical series, tests for convergence of series of constant sign

Directly prove the convergence of the following series and find their sums:

$1-\dfrac {1}{2}+\dfrac {1}{4}-\dfrac {1}{8}+\cdots +\dfrac {(-1)^{n-1}}{2^{n-1}}+\cdots .$

Directly prove the convergence of the following series and find their sums: $\left(\dfrac {1}{2}+\dfrac {1}{3}\right)+\left(\dfrac {1}{2^{2}}+\dfrac {1}{3^{2}}\right)+\cdots +\left(\dfrac {1}{2^{n}}+\dfrac {1}{3^{n}}\right)+\cdots .$

Directly prove the convergence of the following series and find their sums: $\dfrac {1}{2}+\dfrac {3}{2^{2}}+\dfrac {5}{2^{3}}+\cdots +\dfrac {2n-1}{2^{n}}+\cdots .$

Directly prove the convergence of the following series and find their sums: $\dfrac {1}{1 \cdot 2}+\dfrac {1}{2 \cdot 3}+\dfrac {1}{3 \cdot 4}+\cdots +\dfrac {1}{n(n+1)}+\cdots .$

Directly prove the convergence of the following series and find their sums: $\dfrac {1}{1 \cdot 4}+\dfrac {1}{4 \cdot 7}+\cdots +\dfrac {1}{(3n-2)(3n+1)}+\cdots .$

Directly prove the convergence of the following series and find their sums:

Directly prove the convergence of the following series and find its sum:$\displaystyle \sum_{n=1}^{\infty }\left(\sqrt{n-2}-2\sqrt{n+1}+\sqrt{n}\right)$.

Investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } \sin nx$.

Prove that if the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ converges, then the series obtained by grouping the terms of this series without changing their order

also converges and has the same sum. The converse is not true. Give an example.

Prove that if the terms of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ are positive, and the series $\displaystyle \sum_{n=1}^{\infty } A_{n}$ obtained by grouping the terms of this series converges, then the original series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ also converges.

Investigate the convergence of the following series: $1-1+1-1+1-1+\cdots .$

Investigate the convergence of the following series: $0.001+\sqrt{0.001}+\sqrt[3]{0.001}+\cdots .$

Investigate the convergence of the following series: $\dfrac {1}{1!}+\dfrac {1}{2!}+\dfrac {1}{3!}+\cdots +\dfrac {1}{n!}+\cdots .$

Investigate the convergence of the following series: $1+\dfrac {1}{3}+\dfrac {1}{5}+\dfrac {1}{7}+\cdots +\dfrac {1}{2n-1}+\cdots .$

Investigate the convergence of the following series: $\dfrac {1}{1001}+\dfrac {1}{2001}+\dfrac {1}{3001}+\cdots +\dfrac {1}{1000n+1}+\cdots .$

Investigate the convergence of the following series: $1+\dfrac {2}{3}+\dfrac {3}{5}+\cdots +\dfrac {n}{2n-1}+\cdots .$

Investigate the convergence of the following series: $1+\dfrac {1}{3^{2}}+\dfrac {1}{5^{2}}+\cdots +\dfrac {1}{(2n-1)^{2}}+\cdots .$

Investigate the convergence of the following series: $\dfrac {1}{\sqrt{2}}+\dfrac {1}{2\sqrt{3}}+\dfrac {1}{3\sqrt{4}}+\cdots +\dfrac {1}{n\sqrt{n+1}}+\cdots .$

Investigate the convergence of the following series: $\dfrac {1}{\sqrt{1 \cdot 3}}+\dfrac {1}{\sqrt{3 \cdot 5}}+\cdots +\dfrac {1}{\sqrt{(2n-1)(2n+1)}}+\cdots .$

Prove that the series formed from the reciprocals of the terms of an arithmetic series is divergent.

Prove that if the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ ($A$) and $\displaystyle \sum_{n=1}^{\infty } b_{n}$ ($B$) both converge and $a_{n} \leq c_{n} \leq b_{n}$ $(n=1,2,\ldots )$, then the series $\displaystyle \sum_{n=1}^{\infty } c_{n}$ ($C$) also converges. If the series ($A$) and ($B$) both diverge, what can be said about the convergence of series ($C$)?

Suppose the two divergent series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ and $\displaystyle \sum_{n=1}^{\infty } b_{n}$ have terms that are nonnegative. What can be said about the convergence of the following two series:

Prove that if the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ $(a_{n} \geq 0)$ converges, then the series $\displaystyle \sum_{n=1}^{\infty } a_{n}^{2}$ also converges. The converse does not hold; give an example.

Prove that if the series $\displaystyle \sum_{n=1}^{\infty } a_{n}^{2}$ and $\displaystyle \sum_{n=1}^{\infty } b_{n}^{2}$ converge, then the series

also converge.

Prove that if

then the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ diverges.

Prove that if the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, whose terms are positive and monotonically decreasing, converges, then

If for $p=1,2,3,\ldots$, $\displaystyle \lim_{n \rightarrow \infty }(a_{n+1}+a_{n+2}+\cdots +a_{n+p})=0$. Does the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ converge?

Using the Cauchy criterion, prove the convergence of the following series of positive terms: $a_{0}+\dfrac {a_{1}}{10}+\cdots +\dfrac {a_{n}}{10^{n}}+\cdots \quad (\left|a_{n}\right|<10)$.

Using the Cauchy criterion, prove the convergence of the following series of positive terms: $\dfrac {\sin x}{2}+\dfrac {\sin 2x}{2^{2}}+\cdots +\dfrac {\sin nx}{2^{n}}+\cdots .$

Using the Cauchy criterion, prove the convergence of the following series of positive terms: $\dfrac {\cos x-\cos 2x}{1}+\dfrac {\cos 2x-\cos 3x}{2}+\cdots +\dfrac {\cos nx-\cos (n+1)x}{n}+\cdots .$

Using the Cauchy criterion, prove the divergence of the following series: $1+\dfrac {1}{2}+\dfrac {1}{3}+\cdots +\dfrac {1}{n}+\cdots .$

Using the Cauchy criterion, prove the divergence of the following series: $1+\dfrac {1}{2}-\dfrac {1}{3}+\dfrac {1}{4}+\dfrac {1}{5}-\dfrac {1}{6}+\cdots .$

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series: $\dfrac {1000}{1!}+\dfrac {1000^{2}}{2!}+\dfrac {1000^{3}}{3!}+\cdots +\dfrac {1000^{n}}{n!}+\cdots .$

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series: $\dfrac {(1!)^{2}}{2!}+\dfrac {(2!)^{2}}{4!}+\cdots +\dfrac {(n!)^{2}}{(2n)!}+\cdots .$

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series: $\dfrac {1!}{1}+\dfrac {2!}{2^{2}}+\dfrac {3!}{3^{3}}+\cdots +\dfrac {n!}{n^{n}}+\cdots .$

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\dfrac {(1!)^{2}}{2}+\dfrac {(2!)^{2}}{2^{4}}+\dfrac {(3!)^{2}}{2^{9}}+\cdots +\dfrac {(n!)^{2}}{2^{n^{2}}}+\cdots .$

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\dfrac {1000}{1}+\dfrac {1000 \cdot 1001}{1 \cdot 3}+\dfrac {1000 \cdot 1001 \cdot 1002}{1 \cdot 3 \cdot 5}+\cdots .$

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\dfrac {4}{2}+\dfrac {4 \cdot 7}{2 \cdot 6}+\dfrac {4 \cdot 7 \cdot 10}{2 \cdot 6 \cdot 10}+\cdots .$

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty }\left(\sqrt{2}-\sqrt[3]{2}\right)\left(\sqrt{2}-\sqrt[5]{2}\right)\cdots \left(\sqrt{2}-\sqrt[2n+1]{2}\right)$.

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{n^{2}}{\left(2+\frac{1}{n}\right)^{n}}$.

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{n^{n-\frac{1}{n}}}{\left(n+\frac{1}{n}\right)^{n}}$.

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\displaystyle \sum_{n=2}^{\infty } \frac{1}{\sqrt[n]{\ln n}}$.

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{n^{n-1}}{(2n^{2}+n+1)^{\frac{n+1}{2}}}$.

Using the comparison test, d'Alembert's test or Cauchy's test, investigate the convergence of the following series:$\sqrt{2}+\sqrt{2-\sqrt{2}}+\sqrt{2-\sqrt{2+\sqrt{2}}}+\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}+\cdots .$

Prove: if

then $a_{n}=o(q_{1}^{n})$, where $q_{1}>q$.

Prove: if

then the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ converges.

The converse statement is not true. Study the example

Prove that if, for the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ $(a_{n}>0)$, the limit

exists, then

also exists.

The converse statement is not true: if limit (Б) exists, then limit (A) may fail to exist. Study the example

Prove that if

then (a) when $q<1$ the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ converges; (b) when $q>1$ this series diverges (a generalization of Cauchy's test).

Investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{2+(-1)^{n}}{2^{n}}$.

Investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{n\cos^{2}\dfrac {n\pi }{3}}{2^{n}}$.

Investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{n^{3}\left[\sqrt{2}+(-1)^{n}\right]^{n}}{3^{n}}$.

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\left(\dfrac {1}{2}\right)^{p}+\left(\dfrac {1 \cdot 3}{2 \cdot 4}\right)^{p}+\left(\dfrac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^{p}+\cdots .$

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\dfrac {a}{b}+\dfrac {a(a+d)}{b(b+d)}+\dfrac {a(a+d)(a+2d)}{b(b+d)(b+2d)}+\cdots \quad (a>0, b>0, d>0)$.

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{n! e^{n}}{n^{n+p}}$.

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{\sqrt{n!}}{(2+\sqrt{1})(2+\sqrt{2})\cdots (2+\sqrt{n})}$.

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{n! n^{-p}}{q(q+1)\cdots (q+n)} \quad (p>0, q>0)$.

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{p(p+1)\cdots (p+n-1)}{n!} \cdot \frac{1}{n^{q}} \quad (p>0, q>0)$.

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty }\left(\frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}\right)^{p} \cdot \frac{1}{n^{q}}$.

Using Raabe's and Gauss's tests, investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty } \frac{1}{n^{p}}\left(1-\frac{x\ln n}{n}\right)^{n} \quad (p>0)$.

Prove that if $a_{n}>0$ $(n=1,2,\ldots )$ and

then $a_{n}=o\left(\dfrac {1}{n^{p-\varepsilon }}\right)$ $(\varepsilon >0)$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\dfrac {n^{p}+a_{1}n^{p-1}+\cdots +a_{p}}{n^{q}+b_{1}n^{q-1}+\cdots +b_{q}}$, and $n^{q}+b_{1}n^{q-1}+\cdots +b_{q}>0$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\dfrac {1}{n^{p}}\sin \dfrac {\pi }{n}$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\left(\sqrt{n+1}-\sqrt{n}\right)^{p}\ln \dfrac {n-1}{n+1}$ $(n>1)$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\ln^{p}\left(\sec \dfrac {\pi }{n}\right)$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\lg_{b^{n}}\left(1+\sqrt[n]{a}\right)$ $(a>0, b>0)$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\left[e-\left(1+\dfrac {1}{n}\right)^{n}\right]^{p}$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\dfrac {1}{n^{1+\frac{k}{\ln n}}}$.

Find the order of decrease of the general term $a_{n}$, and hence investigate the convergence of the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$, where: $a_{n}=\dfrac {1}{n^{1+\frac{1}{n}}}$.

Prove that if there exists $\alpha >0$ such that for $n \geq n_{0}$, $\dfrac {\ln \dfrac {1}{a_{n}}}{\ln n} \geq 1+\alpha$ $(a_{n}>0)$, then the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ converges; if from some $n_{0}$ on, $\dfrac {\ln \dfrac {1}{a_{n}}}{\ln n} \leq 1$, then the series diverges.

Investigate the convergence of the series with the following general term: $a_{n}=n^{\ln x}$ $(x>0)$.

Investigate the convergence of the series with the following general term: $a_{n}=\dfrac {1}{(\ln \ln n)^{\ln n}}$ $(n>1)$.

Investigate the convergence of the series with the following general term: $a_{n}=\dfrac {1}{(\ln n)^{\ln \ln n}}$ $(n>1)$.

Using the Cauchy integral test, investigate the convergence of the series with the following general term: $a_{n}=\dfrac {1}{n\ln^{p} n}$.

Using the Cauchy integral test, investigate the convergence of the series with the following general term: $a_{n}=\dfrac {1}{n(\ln n)^{p}(\ln \ln n)^{q}}$ $(n>2)$.

Investigate the convergence of the series

Prove that if the terms of the positive series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ decrease monotonically, then the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ and the series $\displaystyle \sum_{n=1}^{\infty } 2^{n} a_{2^{n}}$ converge or diverge together.

Let $f(x)$ be a monotonically nonincreasing positive-valued function. Prove that if the series $\displaystyle \sum_{n=1}^{\infty } f(n)$ converges, then for its remainder $\displaystyle R_{n}=\sum_{k=n+1}^{\infty } f(k)$ we have the following estimate:

Using this formula, find the sum of the series $\displaystyle \sum_{n=1}^{\infty } \frac{1}{n^{3}}$ to within $0.01$.

Prove Yermakov's test: let $f(x)$ be a monotonically decreasing positive-valued function, and let

If $\lambda <1$, then the series $\displaystyle \sum_{n=1}^{\infty } f(n)$ converges; if $\lambda >1$, then the series $\displaystyle \sum_{n=1}^{\infty } f(n)$ diverges.

Prove the Lobachevsky test: let the terms of the positive series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ decrease monotonically to zero; then the series $\displaystyle \sum_{n=1}^{\infty } a_{n}$ and the series

converge or diverge together, where $p_{m}$ is the largest index of the terms $a_{n}$ satisfying the inequality

Investigate the convergence of the following series: $\displaystyle \sum_{n=2}^{\infty } \frac{\sqrt{n+2}-\sqrt{n-2}}{n^{\alpha }}$.

Investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty }\left(\sqrt{n+a}-\sqrt[4]{n^{2}+n+b}\right)$.

Investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty }\left(\ctg \frac{n\pi }{4n-2}-\sin \frac{n\pi }{2n+1}\right)$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty }\left(\frac{1}{\sqrt{n}}-\sqrt{\ln \frac{n+1}{n}}\right)$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{\ln (n!)}{n^{a}}$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } e^{-\sqrt[3]{n}}$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } n^{2} e^{-\sqrt{n}}$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty }\left(n^{\frac{1}{n^{2}+1}}-1\right)$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } e^{\frac{a\ln n+b}{c\ln n+d}}$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{1}{\ln^{2}\left(\sin \dfrac {1}{n}\right)}$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty }\left(\cos \frac{a}{n}\right)^{n^{3}}$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \ln \left(\dfrac {\ch \dfrac {\pi }{n}}{\cos \dfrac {\pi }{n}}\right)$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{n!}{n^{\sqrt{n}}}$.

Investigate the convergence of the following series: $\displaystyle \sum_{n=2}^{\infty } \frac{n^{\ln n}}{(\ln n)^{n}}$.

Investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty }\left(a^{\frac{1}{n}}-\frac{b^{\frac{1}{n}}+c^{\frac{1}{n}}}{2}\right)$ $(a>0, b>0, c>0)$.

Investigate the convergence of the following series: $\displaystyle \sum_{n=1}^{\infty }\left(n^{n^{a}}-1\right)$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty }\left[\ln \frac{1}{n^{a}}-\ln \left(\sin \frac{1}{n^{a}}\right)\right]$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } a^{-(b\ln n+c\ln^{2} n)} \quad (a>0)$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{n^{2n}}{(n+a)^{n+b}(n+b)^{n+a}} \quad (a>0, b>0)$.

Investigate the convergence of the following series:$\displaystyle \sum_{n=1}^{\infty } \frac{\left[(n+1)!\right]^{n}}{2! \cdot 4! \cdots (2n)!}$.

Study the convergence of the series $\displaystyle \sum_{n=1}^{\infty } u_{n}$, whose general term is: $u_{n}=\displaystyle \int_{0}^{\frac{1}{n}} \frac{\sqrt{x}\, dx}{1+x^{2}}$.

Study the convergence of the series $\displaystyle \sum_{n=1}^{\infty } u_{n}$, whose general term is: $u_{n}=\dfrac {1}{\displaystyle \int_{0}^{n} \sqrt[4]{1+x^{4}}\, dx}$.

Study the convergence of the series $\displaystyle \sum_{n=1}^{\infty } u_{n}$, whose general term is: $u_{n}=\displaystyle \int_{n\pi }^{(n+1)\pi } \frac{\sin^{2} x}{x}\, dx$.

Study the convergence of the series $\displaystyle \sum_{n=1}^{\infty } u_{n}$, whose general term is: $u_{n}=\displaystyle \int_{n}^{n+1} e^{-\sqrt{x}}\, dx$.

Study the convergence of the series $\displaystyle \sum_{n=1}^{\infty } u_{n}$, whose general term is: $u_{n}=\displaystyle \int_{0}^{\frac{\pi }{n}} \frac{\sin^{3} x}{1+x}\, dx$.

Study the convergence of the series $\displaystyle \sum_{n=1}^{\infty } u_{n}$, whose general term is: $u_{n}=\dfrac {1!+2!+\cdots +n!}{(2n)!}$.

Study the convergence of the series $\displaystyle \sum_{n=1}^{\infty } u_{n}$, whose general term is: $u_{n}=\dfrac {\displaystyle \sum_{k=1}^{n} \ln^{2} k}{n^{a}}$.

Replace the sequence $x_{n}$ $(n=1,2,\ldots )$ by the corresponding series, and then study its convergence, where: $x_{n}=1+\dfrac {1}{\sqrt{2}}+\cdots +\dfrac {1}{\sqrt{n}}-2\sqrt{n}$.

Replace the sequence $x_{n}$ $(n=1,2,\ldots )$ by the corresponding series, and then study its convergence, where: $x_{n}=\displaystyle \sum_{k=1}^{n} \frac{\ln k}{k}-\frac{(\ln n)^{2}}{2}$.

If