91影视

Let \(\sum_{n=1}^{\infty} a_{n}\) be a series with nonzero terms. Let $$ \rho=\lim _{n \rightarrow \infty}\left|\frac{a_{n+1}}{a_{n}}\right| $$ i. If \(0 \leq \rho<1\), then \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely. ii. If \(\rho>1\) or \(\rho=\infty\), then \(\sum_{n=1}^{\infty} a_{n}\) diverges. iii. If \(\rho=1\), the test does not provide any information.

If \(\lim n \rightarrow \infty a_{n}=c \neq 0\) or \(\lim n \rightarrow \infty a_{n}\) does not exist, then the series \(\sum_{n=1}^{\infty} a_{n}\) diverges.

Using sigma notation, write the following expressions as infinite series. $$ 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $$

Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{n}=1+(-1)^{n} \text { for } n \geq 1 $$

Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} a_{n} \text { where } a_{n}=\frac{2}{n(n+1)} $$

Convergence of Alternating Series For each of the following alternating series, determine whether the series converges or diverges. a. \(\sum_{n=1}^{\infty}(-1)^{n+1} / n^{2}\) b. \(\sum_{n=1}^{\infty}(-1)^{n+1} n /(n+1)\)

Use the comparison test to determine whether the following series converge. $$ \sum_{n=1}^{\infty} a_{n} \text { where } a_{n}=\frac{1}{n(n+1 / 2)} $$

Find the first six terms of each of the following sequences, starting with \(n=1\). $$ a_{n}=n^{2}-1 \text { for } n \geq 1 $$

For each of the following series, use the ratio test to determine whether the series converges or diverges. a. \(\sum_{n=1}^{\infty} \frac{2^{n^{n}}}{n !}\) b. \(\sum_{n=1}^{\infty} \frac{n^{n}}{n !} \sum_{n=1}^{\infty} \frac{(-1)^{n}(n !)^{2}}{(2 n) !}\) c. \(\sum_{n=1}^{\infty} \frac{(-1)^{n}(n !)^{2}}{(2 n) !}\)

Using sigma notation, write the following expressions as infinite series. $$ 1-1+1-1+\cdots $$