91影视

State whether the given series converges and explain why. $$ 1+\frac{\pi}{e}+\frac{\pi^{2}}{e^{4}}+\frac{\pi^{3}}{e^{6}}+\frac{\pi^{4}}{e^{8}}+\cdots $$

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

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

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. \(a_{n}=\frac{(\ln n)^{2}}{\sqrt{n}}\)

State whether the given \(p\) -series converges. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\)

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{k}=\frac{\pi^{k}}{k^{2}} $$

Suppose that \(\lim _{n \rightarrow \infty} a_{n}=1, \lim _{n \rightarrow \infty} b_{n}=-1\), and \(0<-b_{n}

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

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

State whether the given series converges and explain why. $$ 1-\sqrt{\frac{\pi}{3}}+\sqrt{\frac{\pi^{2}}{9}}-\sqrt{\frac{\pi^{3}}{27}}+\cdots $$