True or False The series $\sum\limits_{k=1}^{\infty} \dfrac{1}{k^{p}}$ converges if $p\geq 1.$

True or False According to the Test for Divergence, an infinite series $\sum\limits_{k=1}^{\infty }a_{k}$ converges if $\lim\limits_{n\,\rightarrow \,\infty \,}\,a_{n}=0.$

True or False If a series is absolutely convergent, then it is convergent.

True or False If a series is not absolutely convergent, then it is divergent.

True or False According to the Ratio Test, a series $\sum\limits_{k=1}^{\infty }a_{k}$ of nonzero terms converges if $\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert <1.$

To use the Comparison Test for Convergence to show that a series $\sum\limits_{k\,=\,1}^{\infty }a_{k}$ of positive terms converges,find a series $\sum\limits_{k\,=\,1}^{\infty }b_{k}$ that is known to converge and show that 0 < _______ ≤ ______ .

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{9k^{3}+5k^{2}}{k^{5/2}+4}$

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

$6+2+\dfrac{2}{3}+\dfrac{2}{9}+\dfrac{2}{27}+\cdots$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{1}{k^{2}}\sin \dfrac{\pi }{k}$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{3k+2}{k^{3}+1}$

$1+\dfrac{2^{2}+1}{2^{3}+1}+\dfrac{3^{2}+1}{3^{3}+1}+\dfrac{4^{2}+1}{4^{3}+1}+\cdots$

$\sum\limits_{k=1}^{\infty }\dfrac{k+4}{k\sqrt{3k-2}}$

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

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{3^{2k-1}}{k^{2}+2k}$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{5^{k}}{k!}$

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

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{k^{2}+4}{e^{k}}$

$\dfrac{2}{3}-\dfrac{3}{4}\cdot \dfrac{1}{2}+\dfrac{4}{5}\cdot \dfrac{1}{3}-\dfrac{5}{6}\cdot \dfrac{1}{4}+\cdots$

$2+\dfrac{3}{2}\cdot \dfrac{1}{4}+\dfrac{4}{3}\cdot \dfrac{1}{4^{2}}+\dfrac{5}{4}\cdot \dfrac{1}{4^{3}}+\cdots$

$1+\dfrac{1\cdot 3}{2!}+\dfrac{1\cdot 3\cdot 5}{3!}+\dfrac{1\cdot 3\cdot 5\cdot 7}{4!}+\cdots$

$\dfrac{1}{\sqrt{1\cdot 2\cdot 3}}+\dfrac{1}{\sqrt{2\cdot 3\cdot 4}}+\dfrac{1}{\sqrt{3\cdot 4\cdot 5}}+\cdots$

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

$\sum\limits_{k\,=\,1}^{\infty}k^{3}e^{-k^{4}}$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{1}{\sqrt{k}+100}$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{k^{2}+5k}{3+5k^{2}}$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{1}{\sqrt[3]{k^{4}+4}}$

$\sum\limits_{k=1}^{\infty }\dfrac{1}{11}\left(\dfrac{-3}{2}\right) ^{\!\!k}$

$\dfrac{1}{3}-\dfrac{2}{4}+\dfrac{3}{5}-\dfrac{4}{6}+\cdots$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{k(-4)^{3k}}{5^{k}}$

$\sum\limits_{k\,=\,1}^{\infty}\left( -\dfrac{1}{k}\right) ^{\!\!k}$

$\sum\limits_{k\,=\,1}^{\infty}\dfrac{5}{2^{k}+1}$

$\sum\limits_{k\,=\,1}^{\infty }e^{-k^{2}}$

$\dfrac{\sin \sqrt{1}}{1^{3/2}}+\dfrac{\sin \sqrt{2}}{2^{3/2}}+\dfrac{\sin \sqrt{3}}{3^{3/2}}+\cdots$

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

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

$\sum\limits_{k\,=\,2}^{\infty}\left( \dfrac{\ln k}{1000}\right) ^{\!\!k}$

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

$\sum\limits_{k\,=\,1}^{\infty }\dfrac{\tan ^{-1}k}{k^{2}}$

$\sum\limits_{k\,=\,1}^{\infty }\left(\sqrt{k+1}-\sqrt{k}\right)$

$\sum\limits_{k\,=\,4}^{\infty }\left(\dfrac{1}{k-3}-\dfrac{1}{k}\right)$

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

Determine whether $1+\dfrac{1\cdot 2}{1\cdot 3}+\dfrac{1\cdot 2\cdot 3}{1\cdot 3\cdot 5}+\dfrac{1\cdot 2\cdot 3\cdot 4}{1\cdot 3\cdot 5\cdot 7}+\cdots$ converges or diverges.

$\sum\limits_{k\,=\,1}^{\infty }\dfrac{\ln k}{2k^{3}-1}$

$\sum\limits_{k\,=\,1}^{\infty }\sin ^{3}\left(\dfrac{1}{k}\right)$