Infinite Series Tests

 $\sum_{n=0}^{\infty}\frac{1}{n}$  

• Converges

• Diverges

 $\sum_{n=1}^{\infty}\frac{1}{n^2}$  

• converges

• diverges

Which series is equivalent to

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

•  $\sum_{n=0}^{\infty}\left(n+1\right)\left(\frac{1}{2}\right)^{n-2}$  

•  $\sum_{n=0}^{\infty}\left(n\right)\left(\frac{1}{2}\right)^{n-1}$  

•  $\sum_{n=0}^{\infty}\left(n+2\right)\left(\frac{1}{2}\right)^{n-1}$  

•  $\sum_{n=0}^{\infty}\left(n\right)\left(\frac{1}{2}\right)n-3$  

Find the sum

 $\sum_{n=0}^{\infty}5\left(\frac{1}{3}\right)^n$  

• 15/2

• 15

• 10/3

• Series Diverges

Find the sum

 $\sum_{n=1}^{\infty}4\left(\frac{2}{5}\right)^n$  

• 20/3

• 5/3

• 10/3

• Series 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

 $\sum_{n=1}^{\infty}\left(-1\right)^n\ 5n$  

• Converges by Alternating Series Test

• Diverges by Alternating Series Test

• Converges by n-th term test

• Diverges by n-th term test

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

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

if

 $\sum_{n=1}^{\infty}a_n$  converges and  $\lim_{n\rightarrow\infty}\left(\frac{a_n}{b_n}\right)=5$  

•  $\sum_{n=1}^{\infty}b_n$  also converges

•  $\sum_{n=1}^{\infty}b_n$  diverges

•  $\sum_{n=1}^{\infty}\frac{a_n}{b_n}_{ }$  converges

•  $\sum_{n=1}^{\infty}\frac{a_n}{b_n}$  diverges