91Ó°ÊÓ

Find a power series representation for the function and determine the radius of convergence. $$ f(x)=\frac{x}{(1+4 x)^{2}} $$

Find the Maclaurin series for \(f(x)\) using the definition of a Maclaurin series. [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. $$ f(x)=\sinh x $$

Use the Ratio Test to determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{\cos (n \pi / 3)}{n !} $$

Test the series for convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot(2 n-1)}{2 \cdot 5 \cdot 8 \cdot \cdots \cdot(3 n-1)} $$

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

Determine whether the series is convergent or divergent. $$ \sum_{n=1}^{\infty} \frac{1}{n^{2}+4} $$

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

Find the radius of convergence and interval of convergence of the series. $$ \sum_{n=1}^{\infty} \frac{\sqrt{n}}{8^{n}}(x+6)^{n} $$

Find the Maclaurin series for \(f(x)\) using the definition of a Maclaurin series. [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. $$ f(x)=\cosh x $$

Test the series for convergence or divergence. $$ \sum_{n=2}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}-1} $$