91Ó°ÊÓ

Find the interval of convergence of the series. $$ \sum_{n=2}^{\infty} \frac{1}{n(\ln n)^{2}}(x-1)^{n} $$

If the interval of convergence of \(\sum a_{n} x^{n}\) is \((-r, r],\) prove that the series is conditionally convergent at \(r\).

If \(\sum a_{n}\) and \(\sum b_{n}\) are both convergent series, is \(\sum a_{n} b_{n}\) convergent? Explain.

Use Maclaurin's formula with remainder to establish the approximation formula, and state, in terms of decimal places, the accuracy of the approximation if \(|x| \leq 0.1\). $$ \ln (1+x) \approx x-\frac{x^{2}}{2}+\frac{x^{3}}{3} $$

Use known convergent or divergent series, together with Theorem (11.20) or \((11.21),\) to determine whether the series is convergent or divergent; if it converges, find its sum. $$ \sum_{n=1}^{\infty}\left[\frac{1}{8^{n}}+\frac{1}{n(n+1)}\right] $$

Terms of the sequence defined recursively by \(a_{1}=5\) and \(a_{k+1}=\sqrt{a_{k}}\) may be generated by entering 5 and pressing \(\sqrt{\sqrt{x}}\) repeatedly. (a) Describe what happens to the terms of the sequence as \(k\) increases. (b) Show that \(a_{n}=5^{1 / 2^{n}}\), and find \(\lim _{n \rightarrow \infty} a_{n}\).

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty} \frac{\ln n}{n^{3}} $$

Find the radius of convergence of the series. $$ \sum_{n=0}^{\infty} \frac{1}{(n+5) !}(x+5)^{n} $$

Use Maclaurin's formula with remainder to establish the approximation formula, and state, in terms of decimal places, the accuracy of the approximation if \(|x| \leq 0.1\). $$ \cosh x \approx 1+\frac{x^{2}}{2} $$

Determine whether the series converges or diverges. $$ \sum_{n=1}^{x} \frac{\sin n+2^{n}}{n+5^{n}} $$