91Ó°ÊÓ

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ a_{n}=\frac{\cos ^{2} n}{2^{n}} $$

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ a_{n}=\sqrt[n]{2^{1+3 n}} $$

Determine whether the series is convergent or divergent by expressing \(s_{n}\) as a telescoping sum (as in Example 8 ). If it is convergent, find its sum. $$ \sum_{n=1}^{\infty} \ln \frac{n}{n+1} $$

Show that if \(a_{n}>0\) and \(\Sigma a_{n}\) is convergent, then \(\Sigma \ln \left(1+a_{n}\right)\) is convergent.

For which positive integers \(k\) is the following series convergent? $$ \sum_{n=1}^{\infty} \frac{(n !)^{2}}{(k n) !} $$

(a) Show that \(\Sigma_{n=1}^{\infty} x^{n} / n !\) converges for all \(x .\) (b) Deduce that \(\lim _{n \rightarrow \infty} x^{n} / n !=0\) for all \(x .\)

Determine whether the series is convergent or divergent by expressing \(s_{n}\) as a telescoping sum (as in Example 8 ). If it is convergent, find its sum. $$ \sum_{n=1}^{\infty} \frac{3}{n(n+3)} $$

Find the Maclaurin series of \(f\) (by any method) and its radius of convergence. Graph \(f\) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \(f ?\) $$ f(x)=\cos \left(x^{2}\right) $$

Find all positive values of \(b\) for which the series \(\sum_{n=1}^{\infty} b^{\ln n}\) converges.

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ a_{n}=n \sin (1 / n) $$