Convergence of a Series with Exponential Terms

[Mattofix]

Homework Statement

The series is shown in the answer

The Attempt at a Solution

http://aycu24.webshots.com/image/43903/2005459568713576239_rs.jpg

 

[Mattofix]

i feel like I am going in the totally wrong direction - if so please can someone please point me the right way.

 

[Dick]

Expand the denominator first. The n^2 terms cancel so the denominator is basically proportional to n. The numerator, I think, tends to 1. Do you agree? So do you think it converges or diverges? It's good to have opinion to organize the strategy before you start trying to prove anything.

 

[rootX]

I am getting:

1+exp(-n)
----------
4n.e^n

Now, using ratio test I can prove that 1/(4n.e^n) is converging, and I also know that 1+exp(-n) is converging so that means there product is converging?

My Second approach

lim n--> inf (this series)/ [(1+exp(n))/exp(n)] is equal to 0, so this is equivalent to:
exp(n)+1
----------
exp(n)

now this seems wrong because my series does not converge. What have done wrong here?

 

[Dick]

The nth term in the series is (1+e^(-n))/((n+1)^2-(n-1)^2). The denominator is 4n. The numerator converges to one. It's begging for a comparison test.

 

[Mattofix]

thanks.

(1+exp(-n))/4n > 1/4n

and since 1/4n diverges so must the series.