91影视

Prove that if $\underset{k=1}{\overset{鈭�}{鈭�}}{a}_k$is a convergent series with $a_k鈮�0$for every positive integer k, then the series $\underset{k=1}{\overset{鈭�}{鈭�}}{a}_k^2$converges.

In Exercises 51鈥�54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{k}{k+2}\right\}$

Determine whether the series $\underset{k=0}{\overset{鈭�}{鈭�}}{\left(\frac{-9}{8}\right)}^k$converges or diverges. Give the sum of the convergent series.

Let $a:[1,鈭�)鈫�\mathrm{鈩�}$be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral ${鈭�}_1^{鈭�}a\left(x\right)dx$converges, then the series localid="1649180069308" $\underset{\mathrm{k}=1}{\overset{鈭�}{鈭�}}a\left(k\right)$does too.

Determine whether the sequence converges or diverges. If the sequence converges, give the limit.

$\left\{k^{1/k!}\right\}$

In Exercises 51鈥�54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{1}{k!}\right\}$

Use any convergence test from Sections 7.4鈥�7.6 to determine whether the series in Exercises 41鈥�59 converge or diverge. Explain why each series that meets the hypotheses of the test you select does so.

$\underset{k=0}{\overset{鈭�}{鈭�}}\frac{2路5路8路路路\left(3k+2\right)}{1路5路9路路路\left(4k+1\right)}$

Prove that if $\underset{k=1}{\overset{鈭�}{鈭�}}{a}_k$is a convergent series with $a_k鈮�0$for every positive integer k, then the series $\underset{k=1}{\overset{鈭�}{鈭�}}\frac{a_k}{k}$converges.

In Exercises 52鈥�57, do each of the following:

(a) Show that the given alternating series converges.

(b) Compute $$S_{10}$$ and use Theorem 7.38 to find an interval containing the sum $$L$$ of the series.

(c) Find the smallest value of $$n$$ such that Theorem 7.38 guarantees that $$S_{n}$$ is within $$10^{鈭�6}$$ of $$L$$.

\[ \sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)!} \]

Determine whether the series $\underset{k=0}{\overset{鈭�}{鈭�}}\frac{2^{k+2}}{5^{k-1}}$converges or diverges. Give the sum of the convergent series.