91影视

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

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

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

Find all values of \(\boldsymbol{p}\) and \(q\) such that \(\sum_{n=1}^{\infty} \frac{n^{p}}{(n !)^{q}}\) converges.

Evaluate the following telescoping series or state whether the series diverges. $$ \sum_{n=1}^{\infty} 2^{1 / n}-2^{1 /(n+1)} $$

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. $$ n^{1 / n}, n \geq 3 $$

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. $$ a_{k}=\left(\frac{1}{k+1}+\frac{1}{k+2}+\cdots+\frac{1}{3 k}\right)^{k} $$

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. $$ n^{-1 / n}, n \geq 3 $$

Evaluate the following telescoping series or state whether the series diverges. $$ \sum_{n=1}^{\infty} \frac{1}{n^{13}}-\frac{1}{(n+1)^{12}} $$

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. $$ a_{n}=\left(n^{1 / n}-1\right)^{n} $$