91Ó°ÊÓ

In Exercises 52–57, 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 $$10^{−6}$$ of $$L$$.

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

Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work.

$\underset{k→∞}{\lim }\left(1+\frac{1}{k}\right)\left(2-\frac{1}{k}\right)$

In Exercises 56 and 57 we ask you to complete the proof of Theorem 7.33. For these exercises let $\underset{k=1}{\overset{∞}{∑}}{a}_k$and $\underset{k=1}{\overset{∞}{∑}}{b}_k$be two series with positive terms.

Show that if $\underset{k→∞}{\lim }\frac{a_k}{b_k}=∞$and $\underset{k=1}{\overset{∞}{∑}}{b}_k$diverges, then $\underset{k=1}{\overset{∞}{∑}}{a}_k$diverges.

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\sqrt{1-\frac{1}{k}}\right\}$

Determine whether the series $\underset{k=0}{\overset{∞}{∑}}\frac{{\left(-3\right)}^{k+1}}{4^{k-2}}$converges or diverges. Give the sum of the convergent series.

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

$\underset{k=1}{\overset{∞}{∑}}\frac{k!}{1·3·5·7···(2k-1)}$

In Exercises 55– 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

$\left\{\frac{3^k}{2·4·6···\left(2k\right)}\right\}$

Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work.

$\underset{k→∞}{\lim }\frac{k-7}{3k+5}$

Use the divergence test to prove that a geometric series $\underset{k=1}{\overset{∞}{∑}}c{r}^k$diverges when$\left|r\right|â‰\yen 1$ and $c≠0.$

In Exercises 56 and 57 we ask you to complete the proof of Theorem 7.33. For these exercises let $\underset{k=1}{\overset{∞}{∑}}{a}_k$and $\underset{k=1}{\overset{∞}{∑}}{b}_k$be two series with positive terms.

Show that if $\underset{k→∞}{\lim }\frac{a_k}{b_k}=0$and $\underset{k=1}{\overset{∞}{∑}}{b}_k$converges, then localid="1649095537930" $\underset{k=1}{\overset{∞}{∑}}{a}_k$converges.