91Ó°ÊÓ

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ \begin{aligned} a_{n}=&(\ln 3-\ln 2)+(\ln 4-\ln 3)+(\ln 5-\ln 4)+\cdots \\\ &+(\ln (n-1)-\ln (n-2))+(\ln n-\ln (n-1)) \end{aligned} $$

In Exercises \(57 - 82 ,\) use any method to determine whether the series converges or diverges. Give reasons for your answer. $$ \sum _ { n = 2 } ^ { \infty } \frac { 1 } { 1 + 2 + 2 ^ { 2 } + \cdots + 2 ^ { n } } $$

In each of the geometric series in Exercises \(77-80,\) write out the first few terms of the series to find \(a\) and \(r,\) and \(r\) , and find the sum of the series. Then express the inequality \(|r|<1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\ln \left(1+\frac{1}{n}\right)^{n} $$

In each of the geometric series in Exercises \(77-80,\) write out the first few terms of the series to find \(a\) and \(r,\) and \(r\) , and find the sum of the series. Then express the inequality \(|r|<1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$\sum_{n=0}^{\infty}(-1)^{n} x^{2 n}$$

In Exercises \(57 - 82 ,\) use any method to determine whether the series converges or diverges. Give reasons for your answer. $$ \sum _ { n = 2 } ^ { \infty } \frac { 1 + 3 + 3 ^ { 2 } + \cdots + 3 ^ { n - 1 } } { 1 + 2 + 3 + \cdots + n } $$

In Exercises \(57 - 82 ,\) use any method to determine whether the series converges or diverges. Give reasons for your answer. $$ \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { e ^ { n } } { e ^ { n } + e ^ { n ^ { 2 } } } $$

In each of the geometric series in Exercises \(77-80,\) write out the first few terms of the series to find \(a\) and \(r,\) and \(r\) , and find the sum of the series. Then express the inequality \(|r|<1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$\sum_{n=0}^{\infty} 3\left(\frac{x-1}{2}\right)^{n}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(\frac{3 n+1}{3 n-1}\right)^{n} $$

In each of the geometric series in Exercises \(77-80,\) write out the first few terms of the series to find \(a\) and \(r,\) and \(r\) , and find the sum of the series. Then express the inequality \(|r|<1\) in terms of \(x\) and find the values of \(x\) for which the inequality holds and the series converges. $$\sum_{n=0}^{\infty} \frac{(-1)^{n}}{2}\left(\frac{1}{3+\sin x}\right)^{n}$$