Infinite Series Convergence

 $\sum_{n=0}^{\infty}\frac{1}{n}$  

• Converges

• Diverges

 $\sum_{n=1}^{\infty}\frac{1}{n^2}$  

• converges

• diverges

 $\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n$  

• Converges by nth term test

• Diverges by nth term test

• Converges by Comparison to  $\sum_{n=1}^{\infty}\frac{1}{n^2}$  

• Converges by Geometric Series 

 $\sum_{n=1}^{\infty}\frac{1}{n^{1.01}}$  

• Convergent p-series

• Divergent p-series

• Convergent geometric series

• Divergent geometric series

 $\sum_{n=1}^{\infty}\frac{4}{n^3+2}$  

• Diverges by comparison to  $\sum_{n=1}^{\infty}\frac{4}{n^3}$  

• Converges by comparison to  $\sum_{n=1}^{\infty}\frac{4}{n^3}$  

• Diverges by comparison to  $\sum_{n=1}^{\infty}\frac{4}{n}$  

• Converges by comparison to  $\sum_{n=1}^{\infty}\frac{4}{n}$  

Which series diverge? Check all that do, please and thank-you.

•  $\sum_{n=1}^{\infty}\frac{3n^2}{5n-1}$ 

• $\sum_{n=1}^{\infty}\frac{1}{n^2}$

• $\sum_{n=1}^{\infty}\frac{1}{n^{\frac{1}{2}}}$

• $\sum_{n=1}^{\infty}7\left(\frac{4}{5}\right)^n$

 $\sum_{n=1}^{\infty}\frac{1}{n^{2p}+n\ }$  

For what values of p does this series diverge?

•  $p<\frac{1}{2}$  

•  $p\le\frac{1}{2}$  

•  $p>\frac{1}{2}$  

•  $p\ge\frac{1}{2}$  

Which series converges?

• $\sum_{n=1}^{\infty}\frac{99}{n}$

• $\sum_{n=1}^{\infty}\frac{n}{n^3+4}$

• $\sum_{n=1}^{\infty}\frac{3n^2}{n^3+4n}$

• $\sum_{n=1}^{\infty}\frac{2n^4}{n^4+5n^3}$

 $\sum_{n=1}^{\infty}\ \frac{1}{n\left(n+1\right)}$  

• diverges by Direct Comparison to the Harmonic Series 

• converges by Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n}$  

• converges by Direct Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n^2}$ 

• diverges by Limit Comparison to  $\sum_{n=1}^{\infty}\ \frac{1}{n^2}$ 

 $\sum_{n=1}^{\infty}\ \frac{1}{2n-1}$  diverges by

• Direct Comparison to the Harmonic Series

• it is the Harmonic Series

• Direct Comparison to a Geometric Series where  $\left|r\right|=2$  

•  Nth Term Divergence