91Ó°ÊÓ

Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

36.$\underset{k=1}{\overset{∞}{∑}}\frac{k}{k^2+3}$

Find the least upper bound of the sequences in Exercises 37–42

$\left\{2-\frac{1}{k^2}\right\}$

In Exercises 35–40 use the root test to analyze whether the given series converges or diverges. If the root test is inconclusive, use a different test to analyze the series.

$\underset{k=0}{\overset{∞}{∑}}\frac{1}{k^3+1}$

Evaluate the finite sums.

$\underset{k=0}{\overset{100}{∑}} \frac{1}{2^k}$

For each of the sequences in Exercises 23–52 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.

$\left\{\frac{2^k}{k!}\right\}$

Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

37.$\underset{k=1}{\overset{∞}{∑}}\frac{\ln k}{k}$

Use any convergence tests from Sections 7.4–7.7 to determine whether the series in Exercises 36–51 converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.
$\underset{k=2}{\overset{∞}{∑}}\frac{1}{k^2-1}$

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.

$\underset{k=0}{\overset{∞}{∑}}\frac{{(-1)}^k}{3^k}$

Find the least upper bound of the sequences in Exercises 37–42

$\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},…\right\}$

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

$\underset{k=1}{\overset{∞}{∑}}\sin \left(\frac{1}{k^2}\right)$