Chapter 4

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). $$ a_{1}=1 \text { and } a_{n+1}=-a_{n} \text { for } n \geq 1 $$

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

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

Use the limit comparison test to determine whether each of the following series converges or diverges. $$ \sum_{n=1}^{\infty}\left(\frac{\ln n}{n^{0.6}}\right)^{2} $$

For each of the following sequences, if the divergence test applies, either state that \(\lim _{n \rightarrow \infty} a_{n}\) does not exist or find \(\lim _{n \rightarrow \infty} a_{n} .\) If the divergence test does not apply, state why. \(a_{n}=\frac{(2 n+1)(n-1)}{(n+1)^{2}}\)

Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{(2 n) !}{n^{2 n}} $$

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). $$ a_{1}=2 \text { and } a_{n+1}=2 a_{n} \text { for } n \geq 1 $$

For each of the following series, use the sequence of partial sums to determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} $$

Use the ratio test to determine whether \(\sum_{n=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is given in the following problems. State if the ratio test is inconclusive. $$ \sum_{n=1}^{\infty} \frac{(2 n) !}{(2 n)^{n}} $$

Find a function \(f(n)\) that identifies the \(n\) th term \(a_{n}\) of the following recursively defined sequences, as \(a_{n}=f(n)\). $$ a_{1}=1 \text { and } a_{n+1}=(n+1) a_{n} \text { for } n \geq 1 $$