# How do you use the ratio test to test the convergence of the series #∑3^k/((k+1)!)# from n=1 to infinity?

##### 1 Answer

Oct 14, 2015

The series converges absolutely.

#### Explanation:

The ratio test states the following:

- Consider two consecutive terms
#a_k# and#a_{k+1}# ; - Divide the latter by the former and consider the absolute value:
#abs(a_{k+1}/a_k)# ; - Try to compute the limit of this ratio:
#lim_{k\to\infty}abs(a_{k+1}/a_k)# ;

**THEN**, if the limit exists:

- If it's bigger then
#1# (strictly) , the series does not converge; - If it's smaller then
#1# (strictly), the series converges absolutely; - If it equals one, the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

In your case,

Before dividing, it is useful to consider that:

#3^{k+1}=3*3^k# ,#(k+2)! =(k+2)(k+1)!# , and that- dividing by a fraction means to multiply for the inverse of that fraction.

Now we can divide:

We can simplify a lot of stuff:

Now we can easily take the limit: