Testing Series for Convergence/Divergence

 $\sum_{n=1}^{\infty}\frac{n}{\sqrt{n^5}}$  

• Converges

• Diverges

 $\sum_{n=1}^{\infty}\frac{n+10}{10n+1}$  

• Converges

• Diverges

 $\sum_{n=1}^{\infty}\frac{6}{4^{3n}}$  

• Converges

• Diverges

 $\sum_{n=1}^{\infty}ne^{-n}$  

• Converges

• Diverges

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

• Converges

• Diverges

 $\sum_{n=1}^{\infty}\frac{1}{n^{1.06}}$  

• Converges

• Diverges

 $\sum_{n=1}^{\infty}\frac{1}{4^n}$  

The series above is a

• geometric series

• p-series

Let f be a positive, continuous, decreasing function such that

 $a_n=f\left(n\right)$ .  If  $\sum_{n=1}^{\infty}a_n$  converges to k, which of the following must be true? 

•  $\lim_{n\rightarrow\infty\ }a_n=k$  

•  $\int_1^nf\left(x\right)dx=k$  

•  $\int_1^{\infty}f\left(x\right)dx$     diverges

•  $\int_1^{\infty}f\left(x\right)dx\$    converges

•  $\int_1^{\infty}f\left(x\right)dx=k$  

Which of the following series diverge? (check all that apply)

• $\sum_{n=0}^{\infty}\left(\frac{\sin2}{\pi}\right)^n$

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

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

What is the sum of the converging geometric series below?

 $\sum_{n=1}^{\infty}\frac{7^{n+1}}{10^n}$  

(a)