Convergence of Series: Understanding the Limit Comparison Test

[Sabricd]

Hello,

I need help with problem.
Would you mind solving it so I can compare it with my answer?
$$\sum$$ 1/(2n + 3) where n=1 and goes to infinity.

First of all, I got for an= 1/(2n + 3) and for bn= 1/2n
By using the comparison Test I got:

1/(2n + 3) < (1/2n)

and (1/2n) is a familiar divergent harmonic series. Hence an should also be divergent, but no because bn is larger than an...so the Comparison Test does not determine whether or not the series is divergent, so I have to use the Limit Comparison Test.

Well when I did the Limit Comparison Test I eventually got the limit as n goes to infinity of 2/(2 + (3/n)) which equals 1...and one is more than zero, hence by the limit comparison test either both series converge or diverge...but what do I do now...Please correct me if I'm wrong!

I would deeply appreciate the help!
Thanks! :)

 

[micromass]

Well, comparing to 1/(2n) didnt work. Maybe you'll need to compare it to another series. Maybe make the 2 larger?

 

[Sabricd]

Well I have looked at my notes and previous homework. Is it possible...after your comparison test fails to do the limit comparison and if the answer is greater than 0, and therefore both sequences might or might not converge, to go back and test bn?
For instance the problem above ended up being divergent by following that logic and I seem to find the same logic for all my other homework problems. Am I wrong?

 

[micromass]

I'm sorry, I didn't read your first post completely. You are correct, the limit comparison test works just fine. And you can indeed conclude that the series diverges...

 

[Sabricd]

But what happens if your comparison test fails and you limit comparison test is negative or zero? Does that ever happen?

 

[micromass]

Well, the test can't be negative, since the test is only applicable to series with positive terms.
If you get zero or infinity, then the test is inconclusive, you'll have to try something else then. Maybe try another limit comparison test, but compare with another series...