91Ó°ÊÓ

Find the Taylor series for \(f(x)\) centered at the given value of \(a\). [ Assume that \(f\) has a power series expansion. Do not show that \(\left.R_{n}(x) \rightarrow 0 .\right]\) Also find the associated radius of convergence. $$ f(x)=\sqrt{x}, \quad a=16 $$

Use the Root Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{(-2)^{n}}{n^{n}} $$

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ a_{n}=2+(0.86)^{n} $$

Determine whether the series converges or diverges. $$ \sum_{n=2}^{\infty} \frac{1}{n \sqrt{n^{2}-1}} $$

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

Show that the series is convergent. How many terms of the series do we need to add in order to ind the sum to the indicated accuracy? $$ \sum_{n=1}^{\infty}\left(-\frac{1}{n}\right)^{n} \quad(| \text { error } |<0.00005) $$

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. $$ \sum_{n=1}^{\infty} \frac{6 \cdot 2^{2 n-1}}{3^{n}} $$

Determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{n}{n^{4}+1} $$

Determine whether the sequence converges or diverges. If it converges, find the limit. $$ a_{n}=3^{n} 7^{-n} $$

Determine whether the series converges or diverges. $$ \sum_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{2} e^{-n} $$