91影视

For each of the following sequences, whose \(n\) th terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. $$ \ln \left(1+\frac{1}{n}\right) $$

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1}\left((n+1)^{2}-n^{2}\right)\)

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$ a_{n}=\left(1-\frac{1}{n}\right)^{n^{2}} $$

For which \(r>0\) does the series \(\sum_{n=1}^{\infty} \frac{r^{n^{2}}}{2^{n}}\) converge?

Use the integral test to determine whether the following sums converge. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n+5}}\)

Use the identity \(\frac{1}{1-y}=\sum_{n=0}^{\infty} y^{n}\) to express the function as a geometric series in the indicated term. $$ \frac{1}{1+\sin ^{2} x} \text { in } \sin x $$

For each of the following sequences, whose \(n\) th terms are indicated, state whether the sequence is bounded and whether it is eventually monotone, increasing, or decreasing. $$ \cos \left(n^{2}\right) $$

Use the integral test to determine whether the following sums converge. \(\sum_{n=2}^{\infty} \frac{1}{n \ln n}\)

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1}\left(\frac{1}{n}-\frac{1}{n+1}\right)\)

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$ \left.a_{k}=\left(\frac{1}{k+1}+\frac{1}{k+2}+\cdots+\frac{1}{2 k}\right)^{k} \text { (Hint: Compare } a_{k}^{1 / k} \text { to } \int_{k}^{2 k} \frac{d t}{t} .\right) $$