91影视

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder $R_{10}$.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that.

$\underset{k=2}{\overset{鈭�}{鈭�}}\frac{1}{k{\left(\ln k\right)}^2}$

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.

$\underset{k=1}{\overset{鈭�}{鈭�}}\frac{k^3}{2^k}$

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

$\underset{k=3}{\overset{鈭�}{鈭�}}\frac{k+1}{(k-2{)}^3}$

Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.

$\underset{k=1}{\overset{鈭�}{鈭�}}{\left(-1\right)}^{k}k{e}^{-k^2}$

In Exercises 43鈥�46 give the first five terms for a geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{鈭�}$with the specified values of $candr.$

$c=-1,r=-\frac{1}{2}$

Determine whether the sequence is monotonic or eventually monotonic and whether the sequence is bounded above and/or below. If the sequence converges, give the limit.

$\left\{\tan 鈦�\left(\frac{k蟺}{2}\right)\right\}$

Determine whether the series $\underset{k=0}{\overset{鈭�}{鈭�}}\frac{3}{2^k}$converges or diverges. Give the sum of the convergent series.

For each series in Exercises 44鈥�47, do each of the following:

(a) Use the integral test to show that the series converges.

(b) Use the 10th term in the sequence of partial sums to approximate the sum of the series.

(c) Use Theorem 7.31 to find a bound on the tenth remainder,$R_{10}$.

(d) Use your answers from parts (b) and (c) to find an interval containing the sum of the series.

(e) Find the smallest value of n so that$R_n鈮�{10}^{-6}$

$\underset{k=1}{\overset{鈭�}{鈭�}}\frac{1}{k^2}$

Write each of the arithmetic sequences in Exercises 47鈥�50 in the form $\{c+dk{\}}_{k=0}^{鈭�}$

$-3,7,17,27,鈥�$

Use any convergence test from this section or the previous section to determine whether the series in Exercises 31鈥�48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

$\underset{k=1}{\overset{鈭�}{鈭�}}\frac{1}{2k+7}$