Review of Convergence Tests

nth term test:

If the limit as n approaches infinity of a series is (a)   zero, then the series will diverge.

Geometric Series:

A geometric series will converge if (a)   .

Geometric Series:

If a geometric series converges, the sum of the series will be equal to...

• $\frac{a_i}{r+1}$

• $\frac{a_i}{r-1}$

• $\frac{a_i}{1+r}$

• $\frac{a_i}{1-r}$

Integral test:

If the integral (a)   , then the series converges.

p-series:

A p-series will converge if (a)   .

Direct comparison test:
Given that

 $0<a_n<b_n$  
select which of the following statements is true.

• If  $b_n$  diverges, then  $a_n$  diverges.

• If  $b_n$  converges, then  $a_n$  converges.

• If  $a_n$  converges, then  $b_n$  converges.

• If  $a_n$  diverges, then  $b_n$  converges.

Limit Comparison Test:
Given that

 $\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=L$  , where L is finite and positive, then

•  $a_n$  and  $b_n$  have the same behavior (both converge or both diverge)

•  $a_n$  and  $b_n$  have the opposite behavior (one converges and the other diverges)

Alternating Series Test:
An alternating series will converge if:

 $\lim_{n\rightarrow\infty}\ a_n=$  

(a)  

Alternating Series Test:
An alternating series will converge if:

 $a_n+1$   (a)      $a_n$  

Ratio Test:
A series will converge by the ratio test if

 $\lim_{n\rightarrow\infty}\left|\frac{a_{\left(n+1\right)}}{a_n}\right|$   (a)   1