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

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

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

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

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

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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 < _______ ≤ ______ .

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

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

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\(6+2+\dfrac{2}{3}+\dfrac{2}{9}+\dfrac{2}{27}+\cdots \)

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

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

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\(1+\dfrac{2^{2}+1}{2^{3}+1}+\dfrac{3^{2}+1}{3^{3}+1}+\dfrac{4^{2}+1}{4^{3}+1}+\cdots \)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

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1. Show that the series \(\sum\limits_{k=1}^{\infty }\left[\left(\dfrac{2}{3}\right) ^{\!\!k}-\dfrac{2}{k^{2}+2k}\right]\) converges.
2. Find the sum of the series.

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1. Show that the series \(\sum\limits_{k=1}^{\infty }\left[ \left( -\dfrac{1}{4}\right) ^{\!\!k}+\dfrac{3}{k(k+1) }\right]\) converges.
2. Find the sum of the series.

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1. Determine whether the series \(1-1-\dfrac{1}{2}+ \dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{9}-\dfrac{1}{4}+\dfrac{1}{27}+\dfrac{1}{5 }-\dfrac{1}{81}-\cdots \) converges or diverges.
2. Find the sum of the series if it converges.

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

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