91Ó°ÊÓ

Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{1}{1+2+3+\cdots+n} $$

Which series in Exercises \(53-76\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=0}^{\infty}\left(\frac{1}{\sqrt{2}}\right)^{n}$$

Replace \(x\) by \(-x\) in the Taylor series for \(\ln (1+x)\) to obtain a series for \(\ln (1-x)\) . Then subtract this from the Taylor series for \(\ln (1+x)\) to show that for \(|x|<1\), \begin{equation} \ln \frac{1+x}{1-x}=2\left(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\cdots\right). \end{equation}

In Exercises \(53 - 56 ,\) determine how many terms should be used to estimate the sum of the entire series with an error of less than \(0.001 .\) $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { 1 } { n ^ { 2 } + 3 } $$

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

How many terms of the Taylor series for \(\ln (1+x)\) should you add to be sure of calculating ln \((1.1)\) with an error of magnitude less than \(10^{-8} ?\) Give reasons for your answer.

Which series in Exercises \(53-76\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=0}^{\infty}(\sqrt{2})^{n}$$

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{\ln n}{\ln 2 n} $$

Which of the series in Exercises \(17-56\) converge, and which diverge? Use any method, and give reasons for your answers. $$ \sum_{n=1}^{\infty} \frac{1}{1+2^{2}+3^{2}+\cdots+n^{2}} $$

In Exercises \(53 - 56 ,\) determine how many terms should be used to estimate the sum of the entire series with an error of less than \(0.001 .\) $$ \sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { n } { n ^ { 2 } + 1 } $$