91Ó°ÊÓ

Make up an infinite series of nonzero terms whose sum is a. 1\(\quad\) b. \(-3 \quad\) c. \(0 .\)

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

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(3^{n}+5^{n}\right)^{1 / n} $$

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

Show by example that \(\sum\left(a_{n} / b_{n}\right)\) may diverge even though \(\Sigma a_{n}\) and \(\sum b_{n}\) converge and no \(b_{n}\) equals \(0 .\)

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\tan ^{-1} n $$

Find convergent geometric series \(A=\sum a_{n}\) and \(B=\sum b_{n}\) that illustrate the fact that \(\sum a_{n} b_{n}\) may converge without being equal to \(A B .\)

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\frac{1}{\sqrt{n}} \tan ^{-1} n $$

Show by example that \(\sum _ { n = 1 } ^ { \infty } a _ { n } b _ { n }\) may diverge even if \(\sum _ { n = 1 } ^ { \infty } a _ { n }\) and \(\sum _ { n = 1 } ^ { \infty } b _ { n }\) both converge.

Which of the sequences \(\left\\{a_{n}\right\\}\) converge, and which diverge? Find the limit of each convergent sequence. $$ a_{n}=\left(\frac{1}{3}\right)^{n}+\frac{1}{\sqrt{2^{n}}} $$