Convergence of Series with Alternating Terms

[mvpshaq32]

Homework Statement

E from 1 to infinity [2+(-1)^n]/[n(n^1/2)]

Homework Equations

We only have had learned comparison tests, power series, and divergence tests.

The Attempt at a Solution

I decided to split the function into its numerator multiplied by its denominator:
2+(-1)^n * 1/[n(n^1/2)]
Then I perform the divergence test for the numerator in which the limit doesn't exist, meaning it diverges and the p-series test for the denominator whose ratio is 3/2 which means it converges. I then made the assumption that a diverging function multiplied by a converging function would diverge. But I don't know if I am allowed to do this.

 

[lanedance]

notice that
1 <= [2+(-1)^n] <= 3

 

[vela]

I decided to split the function into its numerator multiplied by its denominator...

I then made the assumption that a diverging function multiplied by a converging function would diverge. But I don't know if I am allowed to do this.

No, you can't do this. You need to look at the term as a whole.

You probably find the numerator confusing. A good tactic is to replace something that looks complicated with a simple quantity that either bounds it from above or from below, depending on what you want to prove. Then show the simplified series converges or diverges. Use lanedance's hint.