91Ó°ÊÓ

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. $$ \sum_{n=1}^{\infty} \frac{1}{(\sqrt{n}+\sqrt{2})^{4}} $$

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

State what conclusion, if any, may be drawn from the Divergence Test. $$ \sum_{n=1}^{\infty} \frac{3^{n}}{3^{n}+4} $$

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

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{10^{n}}{n !}\)

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{3^{2 n}+2}{9^{n+1}+1} $$

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

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n^{100}}{n !}\)

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

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{4+x}\)