91Ó°ÊÓ

Use the Ratio Test to determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{3^{n}}{(n+1)^{n}} $$

Find the Maclaurin series for the function. \(f(x)=\frac{1}{2}\left(e^{x}-e^{-x}\right)=\sinh x\)

Graphical and Numerical Analysis , let \(S_{n}=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\cdots \pm \frac{x^{n}}{n}\) Use a graphing utility to confirm the inequality graphically. Then complete the table to confirm the inequality numerically. $$ S_{2} \leq \ln (x+1) \leq S_{3} $$

Verify that the infinite series converges. \(\sum_{n=0}^{\infty}(0.9)^{n}=1+0.9+0.81+0.729+\cdots\)

Determine the convergence or divergence of the series.\(\sum_{n=1}^{\infty} \frac{(-1)^{n+1} \sqrt{n}}{n+2}\)

Use the Limit Comparison Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \sin \frac{1}{n} $$

Use the Integral Test to determine the convergence or divergence of the \(p\) -series. \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\)

Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. $$ 5,10,20,40, \ldots $$

Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=1}^{\infty} \frac{n}{n+1}(-2 x)^{n-1} $$

Find the \(n\) th Taylor polynomial centered at \(c .\) $$ f(x)=\sqrt{x}, \quad n=4, \quad c=1 $$