91Ó°ÊÓ

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n^{2}}\right)^{n} $$

Writing, explain how to use the geometric series \(g(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}, \quad|x|<1\) to find the series for the function. Do not find the series. $$ f(x)=\frac{5}{1+x} $$

Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. $$ a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right) $$

Determine whether the series converges conditionally or absolutely, or diverges.\(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{(n+1)^{2}}\)

Find the sum of the convergent series. \(\sum_{n=0}^{\infty}\left(\frac{1}{2^{n}}-\frac{1}{3^{n}}\right)\)

In Exercises 47 and 48 , find a Maclaurin series for \(f(x)\). \(f(x)=\int_{0}^{x}\left(e^{-t^{2}}-1\right) d t\)

A friend in your calculus class tells you that the following series converges because the terms are very small and approach 0 rapidly. Is your friend correct? Explain. \(\frac{1}{10,000}+\frac{1}{10,001}+\frac{1}{10,002}+\cdots\)

Find the sum of the convergent series. \(\sum_{n=1}^{\infty}\left[(0.7)^{n}+(0.9)^{n}\right]\)

Find the intervals of convergence of (a) \(f(x)\) (b) \(f^{\prime}(x)\), (c) \(f^{\prime \prime}(x)\), and (d) \(\int f(x) d x .\) Include a check for convergence at the endpoints of the interval. $$ f(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-2)^{n}}{n} $$

Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. $$ a_{n}=1+(-1)^{n} $$