91Ó°ÊÓ

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

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

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

Use the Ratio Test to determine the convergence or divergence of the given series. \(\sum_{n=1}^{\infty} \frac{n !}{11^{n}(n+12)^{13}}\)

Express the given function as a power series in \(x\) with base point \(0 .\) Calculate the radius of convergence \(R\). \(\frac{1}{9-x^{2}}\)

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

Evaluate \(\lim _{n \rightarrow \infty} a_{n}\) for the given sequence \(\left\\{a_{n}\right\\}\). $$ a_{n}=\frac{2^{n}+5^{n}}{7^{n}} $$

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. $$ \sum_{n=1}^{\infty} \frac{2 n-1}{n e^{n}} $$

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

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. $$ \sum_{n=1}^{\infty} \frac{(-4 / 5)^{n}}{n+2} $$