Question

True or False The Ratio Test can be used to show that the series \(\sum\limits_{k\,=\,1}^{\infty }\cos ( k\pi )\) diverges.

Question

True or False In using the Ratio Test, if \( \lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}} \right\vert =L,\) then the sum of the series \(\sum\limits_{k\,=\,1}^{\infty }a_{k}\) equals \(L.\)

Question

True or False In using the Ratio Test, if \( \lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}} \right\vert =1\), then the Ratio Test indicates that the series \( \sum\limits_{k\,=\,1}^{\infty }a_{k}\) converges.

Question

True or False The Root Test works well if the \(n\)th term of a series of nonzero terms involves an \(n\)th root.

Question

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

Question

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

Question

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

Question

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

Question

\(\sum\limits_{k\,=\,1}^{\infty}\dfrac{10^{k}}{(2k)!}\)

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

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

Question

\(\sum\limits_{k\,=\,1}^{\infty}\left(\dfrac{\sqrt{4k^{2}+1}}{k}\right) ^{\!\!k}\)

Question

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

Question

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

Question

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

Question

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

Question

\(\sum\limits_{k\,=\,1}^{\infty}\dfrac{10}{(3k+1)^{k}}\)

Question

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

Question

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

Question

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

Question

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

Question

\(\sum\limits_{k\,=\,1}^{\infty}\left[ \ln \left( e^{3}+\dfrac{1}{k}\right)\right]^{k}\)

Question

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

Question

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

Question

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

Question

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

Question

For the divergent series \(\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{k}\), show that \(\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert =1\).

596

Question

For the convergent series \(\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{ k^{2}}\), show that \(\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{ a_{n+1}}{a_{n}}\right\vert =1\).

Question

Give an example of a convergent series \(\sum\limits_{k\,= \,1}^{\infty }a_{k}\) for which \(\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert \neq \infty \) does not exist.

Question

Give an example of a divergent series \(\sum\limits_{k\,= \,1}^{\infty }a_{k}\) for which \(\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}}\right\vert \neq \infty \) does not exist.

Question

1. Show that the series \(\sum\limits_{k=1}^{\infty }\dfrac{ (-1) ^{k}3^{k}}{k!}\) converges.
2. Use technology to find the sum of the series.

Question

Determine whether the following series is convergent or divergent: \[ \frac{1}{3}-\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}-\frac{4^{3}}{3^{4}} +\cdots +\frac{(-1)^{n-1}n^{3}}{3^{n}}+\cdots \]

Question

Show that \(\lim\limits_{n\,\rightarrow \,\infty }\,\dfrac{n!}{n^{n}}=0\), where \(n\) denotes a positive integer.

Question

Show that the Root Test is inconclusive for \( \sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{k}\) and \(\sum\limits_{k\,= \,1}^{\infty }\dfrac{1}{k^{2}}\).

Question

Use the Ratio Test to find the real numbers \(x\) for which the series \(\sum\limits_{k\,=\,1}^{\infty }\dfrac{x^{k}}{k^{2}}\) converges or diverges.

Question

Prove that \(\sum\limits_{k\,=\,1}^{\infty }a_{k}\) diverges if \(\lim\limits_{n\,\rightarrow \,\infty }\left\vert \dfrac{a_{n+1}}{a_{n}} \right\vert =\infty\).

Question

Prove the Root Test.

Question

The terms of the series \[ \dfrac{1}{4}+\dfrac{1}{2}+\dfrac{1}{8}+\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{8 }+\dfrac{1}{32}+\cdots \]

are \(a_{2k}=\dfrac{1}{2^{k}}\) and \(a_{2k-1}=\dfrac{1}{2^{k+1}}.\)

1. Show that using the Ratio Test to determine whether the series converges is inconclusive.
2. Show that using the Root Test to determine whether the series converges is conclusive.
3. Does the series converge?

Question

Show that the following series converges: \[ 1+\frac{2}{2^{2}}+\frac{3}{3^{3}}+\frac{1}{4^{4}}+\frac{2}{5^{5}}+\frac{3}{ 6^{6}}+\cdots \]

Question

Show that \(\sum\limits_{k\,=\,1}^{\infty }\dfrac{(k+1)^{2}}{(k+2) !}\) converges.

(Hint: Use the Limit Comparison Test and the convergent series \(\sum\limits_{k\,=\,1}^{\infty }\dfrac{1}{k!}.\))

Question

Suppose \(0<a<b<1\). Use the Root Test to show that the series \[ a+b+a^{2}+b^{2}+a^{3}+b^{3}+\cdots \]

converges.

Question

Show that if the Ratio Test indicates a series converges, then so will the Root Test. The converse is not true. Refer to Problem 56.