91Ó°ÊÓ

Use Theorem 2 and, where necessary, limit formula (8.5.1) to calculate the radius of convergence \(R\). Determine the interval of convergence \(I\) by checking the endpoints. $$ \sum_{n=0}^{\infty}\left(\frac{-x}{3}\right)^{n} $$

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. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{5 n-4}{n^{9 / 4}} $$

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{x}{1+x}\)

Use the Comparison Test for Convergence to show that the given series converges. State the series that you use for comparison and the reason for its convergence. $$ \sum_{n=1}^{\infty} \frac{1}{(n+\sqrt{2})^{2}} $$

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{2^{n+1}+5}{2^{n}+3} $$

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{2^{n}}{n^{3}}\)

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. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{2-x}\)

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{n^{2}}{n^{2}+1} $$