91Ó°ÊÓ

A convergent alternating series \(S=\sum_{n=1}^{\infty}(-1)^{n+1} a_{n}\) is given. Use inequality (8.4.1) to find a value of \(N\) such that the partial sum \(S_{N}=\sum_{n=1}^{N}(-1)^{n+1} a_{n}\) approximates the given infinite series to within 0.01 . That is, find \(N\) so that \(\left|S-S_{N}\right| \leq 0.01\) holds. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{n^{2}+15 n} $$

Find the sum of the given series. $$ \sum_{n=1}^{\infty}\left(2^{-n} \cdot 3^{-n}+7^{-n}\right) $$

In each of Exercises \(39-42,\) compute the Taylor polynomial \(T_{3}(x)\) of the given function \(f\) with the given base point \(c\). \(f(x)=(x+5)^{1 / 3} \quad c=3\)

Use the Integral Test to determine whether the given series converges or diverges. Before you apply the test, be sure that the hypotheses are satisfied. $$ \sum_{n=1}^{\infty} \frac{\ln (n)}{n} $$

Use the Limit Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{2^{n}}{1+2^{2 n}} $$

Use the Limit Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \sin (1 / n) $$

Use the Integral Test to determine whether the given series converges or diverges. Before you apply the test, be sure that the hypotheses are satisfied. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}+2 n+1} $$

A convergent alternating series \(S=\sum_{n=1}^{\infty}(-1)^{n+1} a_{n}\) is given. Use inequality (8.4.1) to find a value of \(N\) such that the partial sum \(S_{N}=\sum_{n=1}^{N}(-1)^{n+1} a_{n}\) approximates the given infinite series to within 0.01 . That is, find \(N\) so that \(\left|S-S_{N}\right| \leq 0.01\) holds. $$ \sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{\ln (n+1)} $$

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

Find the sum of the given series. $$ \sum_{n=1}^{\infty}\left(5^{-n} \cdot 3^{-n} \cdot 4^{n}\right) $$