91Ó°ÊÓ

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty} \frac{n}{(\ln n)^{n}} $$

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(n !)^{n}}{\left(n^{n}\right)^{2}} $$

Verify the sum. Then use a graphing utility to approximate the sum with an error of less than \(0.0001\). \(\sum_{n=0}^{\infty}(-1)^{n}\left[\frac{1}{(2 n+1) !}\right]=\sin 1\)

Find the sum of the convergent series. \(\sum_{n=1}^{\infty} \frac{1}{9 n^{2}+3 n-2}\)

Determine the convergence or divergence of the sequence with the given \(n\) th term. If the sequence converges, find its limit. $$ a_{n}=\frac{\sqrt[3]{n}}{\sqrt[3]{n}+1} $$

Let \(f\) be a positive, continuous, and decreasing function for \(x \geq 1\), such that \(a_{n}=f(n)\). Use a graph to rank the following quantities in decreasing order. Explain your reasoning. (a) \(\sum_{n=2}^{7} a_{n}\) (b) \(\int_{1}^{7} f(x) d x\) (c) \(\sum_{n=1}^{6} a_{n}\)

Determine whether the series converges conditionally or absolutely, or diverges.\(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n \sqrt{n}}\)

State the Direct Comparison Test and give an example of its use.

Find the positive values of \(p\) for which the series converges. \(\sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{p}}\)

Verify the sum. Then use a graphing utility to approximate the sum with an error of less than \(0.0001\). \(\sum_{n=0}^{\infty} \frac{2^{n}}{n !}=e^{2}\)