91Ó°ÊÓ

The given series may be shown to converge by using the Alternating Series Test. Show that the hypotheses of the Alternating Series Test are satisfied. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n !} $$

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{\sqrt{n !}}{n^{5} \cdot 7^{n}}\)

A series \(\sum_{n=1}^{\infty} a_{n}\) is given. Calculate the first five partial sums of the series. That is, calculate \(S_{N}=\sum_{n=1}^{N} a_{n}\) for \(N=1,2,3,4,5\). $$ \sum_{n=1}^{\infty}\left(2^{-n}+1\right) $$

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. $$ \text { 9. } \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} $$

Use the Limit Comparison Test to determine whether the given series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{10 n+3}{100 n^{3}-99} $$

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 radius of convergence of the given power series. $$ \sum_{n=0}^{\infty} n(n+1)(n+2)(2 x+3)^{n} $$

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. $$ \sum_{n=3}^{\infty} \frac{1}{n \ln ^{3 / 2}(n)} $$

Write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number. $$ 0.1221212121 \ldots $$

Find the sum of the given series in closed form. State the radius of convergence \(R\). \(\sum_{n=1}^{\infty} n(n+1) x^{n}\)