91Ó°ÊÓ

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{3^{n}} $$

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=1}^{\infty}\left(-\frac{1}{4}\right)^{-k-2} $$

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=\tanh x $$

In Problems 1–6, show that each alternating series converges, and then estimate the error made by using the partial sum as an approximation to the sum S of the series (see Examples 1–3). $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\ln (n+1)} $$

In Problems 1-18, find the terms through \(x^{5}\) in the Maclaurin series for \(f(x)\). Hint: It may be easiest to use known Maclaurin series and then perform multiplications, divisions, and so on. For example, \(\tan x=(\sin x) /(\cos x)\). $$ f(x)=e^{x} \sin x $$

$$ \sum_{n=1}^{\infty} \frac{1}{n \sqrt{n+1}} $$

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} $$

In Problems 1-10, find the power series representation for \(f(x)\) and specify the radius of convergence. Each is somehow related to a geometric series (see Examples 1 and 2). $$ f(x)=\frac{1}{(1-x)^{3}} $$

\(\sum_{k=0}^{\infty} \frac{k}{k^{2}+3}\)

In Problems 1-14, indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series. $$ \sum_{k=0}^{\infty}\left[2\left(\frac{1}{4}\right)^{k}+3\left(-\frac{1}{5}\right)^{k}\right] $$