91Ó°ÊÓ

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

Make up an infinite series of nonzero terms whose sum is $$\begin{array}{llll} \text { a. } & 1 & \text { b. }-3 & \text { c. } 0 \end{array}$$

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

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

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

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

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

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

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