91Ó°ÊÓ

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{2 n}{n+1}\right)^{n} $$

find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}(x-c)^{n}}{n c^{n}} $$

find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=1}^{\infty} \frac{k(k+1)(k+2) \cdot \cdots(k+n-1) x^{n}}{n !}, k \geq 1 $$

Use the power series \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n},|x|<1\) Find the series representation of the function and determine its interval of convergence. $$ f(x)=\frac{1}{(1-x)^{2}} $$

Find the limit (if possible) of the sequence. $$ a_{n}=\frac{2 n}{\sqrt{n^{2}+1}} $$

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=2}^{\infty}\left(\frac{2 n+1}{n-1}\right)^{n} $$

Find the sum of the convergent series. \(\sum_{n=0}^{\infty}\left(\frac{1}{2}\right)^{n}\)

Find the sum of the convergent series. \(\sum_{n=0}^{\infty} 6\left(\frac{4}{5}\right)^{n}\)

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{4 n+3}{2 n-1}\right)^{n} $$

Use the power series \(\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n},|x|<1\) Find the series representation of the function and determine its interval of convergence. $$ f(x)=\frac{x}{(1-x)^{2}} $$