91Ó°ÊÓ

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x\). $$\sum_{n=0}^{\infty}(\ln x)^{n}$$

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

The series in Exercise 5 can also be written as $$ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} \text { and } \sum_{n=1}^{\infty} \frac{1}{(n+3)(n+4)} $$ Write it as a sum beginning with (a) \(n=-2,\) (b) \(n=0\) (c) \(n=5\)

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

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

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

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 \(\Sigma\left(a_{n} / b_{n}\right)\) may diverge even though \(\Sigma a_{n}\) and \(\Sigma b_{s}\) 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. \(a_{n}=\left(\frac{1}{3}\right)^{n}+\frac{1}{\sqrt{2^{n}}}\)

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