91影视

Find the limit of each of the following sequences, using L'H么pital's rule when appropriate. $$ \frac{n^{2}}{2^{n}} $$

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

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n}\left(1-n^{1 / n}\right)\) (Hint: \(n^{1 / n} \approx 1+\ln (n) / n\) for large \(\left.n .\right)\)

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

Use the limit comparison test to determine whether each of the following series converges or diverges. $$ \sum_{n=1}^{\infty}\left(1-e^{-1 / n}\right)\left(\text { Hint: } 1 / e \approx(1-1 / n)^{n}, \text { so } 1-e^{-1 / n} \approx 1 / n .\right) $$

State whether each of the following series converges absolutely, conditionally, or not at all\(\sum_{n=1}^{\infty}(-1)^{n+1} n\left(1-\cos \left(\frac{1}{n}\right)\right)\left(\right.\) Hint \(\cos (1 / n) \approx 1-1 / n^{2}\) for large \(\left.n .\right)\)

State whether the given \(p\) -series converges. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n^{2}}}\)

Find the limit of each of the following sequences, using L'H么pital's rule when appropriate. $$ \frac{(n-1)^{2}}{(n+1)^{2}} $$

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

Find the limit of each of the following sequences, using L'H么pital's rule when appropriate. $$ \frac{\sqrt{n}}{\sqrt{n+1}} $$