91Ó°ÊÓ

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty} \frac{2^{n}+3^{n}}{4^{n}} $$

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty} \frac{2^{n}+4^{n}}{3^{n}+4^{n}} $$

Show that if \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely, then $$\left|\sum_{n=1}^{\infty} a_{n}\right| \leq \sum_{n=1}^{\infty}\left|a_{n}\right|$$

If \(\sum a_{n}\) is a convergent series of positive terms, prove that \(\sum \sin \left(a_{n}\right)\) converges.

Show that neither the Ratio Test nor the Root Test provides information about the convergence of $$\sum_{n=2}^{\infty} \frac{1}{(\ln n)^{p}} \quad(p constant )$$

Which of the sequences \(\left\\{a_{n}\right\\}\) in Exercises \(27-90\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{(-4)^{n}}{n !} $$

Show that if \(\sum_{n=1}^{\infty} a_{n}\) and \(\sum_{n=1}^{\infty} b_{n}\) both converge absolutely, then so do the following. $$\begin{array}{ll}{\text { a. }} & {\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)} & {\text { b. } \sum_{n=1}^{\infty}\left(a_{n}-b_{n}\right)} \\ {\text { c. }} & {\sum_{n=1}^{\infty} k a_{n}(k \text { any number })}\end{array}$$

Let $$a_{n}=\left\\{\begin{array}{ll}{n / 2^{n},} & {\text { if } n \text { is a prime number }} \\ {1 / 2^{n},} & {\text { otherwise. }}\end{array}\right.$$ Does \(\sum a_{n}\) converge? Give reasons for your answer.

Which series in Exercises \(49-68\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$ \sum_{n=1}^{\infty} \ln \left(\frac{n}{n+1}\right) $$

Series for sin \(^{-1} x\) Integrate the binomial series for \(\left(1-x^{2}\right)^{-1 / 2}\) to show that for \(|x|<1\) \begin{equation} \sin ^{-1} x=x+\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot(2 n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)} \frac{x^{2 n+1}}{2 n+1} \end{equation}