# Thread: Infinite series : testing for convergence, exponential involved.

1. ## Infinite series : testing for convergence, exponential involved.

I am asked to see if the summation of (3^n+n)/(4^n-3^n) converge or not.

I have tried ratio test, comparison test and limit comparison test, but none of them work (at least for me provided my limited skills in math)

By comparison test I used 3^n/4^n and 1/(4^n-3^n) for comparison, the same goes for limit comparison (I break the summation up into two part, one with 3^n on top and the other with n on top)

How should I attack the problem? Thank you!

2. Originally Posted by kyva1929
I am asked to see if the summation of (3^n+n)/(4^n-3^n) converge or not.

I have tried ratio test, comparison test and limit comparison test, but none of them work (at least for me provided my limited skills in math)
As these terms are all positive we do not need to take absolute values, but you can imagine they are present if you like:

Divide through top and bottom by $3^n$ to get:

$\frac{3^n+n}{4^n-3^n}=\frac{1+n/3^n}{(4/3)^n-1}$

and for $n$ sufficiently large the right most term is less than $\frac{2}{(4/3)^n/2}$ so:

$\frac{3^n+n}{4^n-3^n}<\frac{4}{(4/3)^n}=4 (3/4)^n$

CB