Convergence of a series by root test

[train449]

Homework Statement

Does the series:

sum from 1 to inf. (n+2)/(n+1)

converge? If so does it converge absolutely?

Homework Equations

Ratio test for series

The Attempt at a Solution

I found this series to converge using the root test, (.jpg attached) however wolfram alpha says otherwise.

 

[vela]

Your work was fine until you said L=3/4. Remember n is going to infinity, so the constant and linear terms don't really matter.

 

[Mark44]

I agree with wolframalpha.
A much simpler test to use is the nth term test for divergence.

 

[train449]

AH Vela you're right. L=1 so the test is inconclusive. I tried the integral test and got infinty so it does diverge, thank you.

 

[train449]

@Mark44 I'm not too sure what the nth term test is

 

[vela]

The n-th term test is the first test you should always try on a series. It's also probably the first one you learned. It says if the terms don't go to 0 in the limit as n goes to infinity, the series diverges.

 

[train449]

Of course! figures I would forget the most basic reasoning. Thank you Vela!

 

[Mark44]

The Nth Term Test for Divergence is simple to use, but in my experience, is often misunderstood by students. If the limit of the nth term of a series isn't zero, the series diverges. This test tells you when a series diverges but it doesn't tell you when a series converges.

 

[train449]

thank you, Mark44, I will make good use of it on my upcoming test!