I need some help with a homework question i'm having difficulty with. Here is the question:
"Use the definition of cauchy sequence to prove that the series $\left(1+\frac{2}{3+1}+\frac{3}{9+1}+\cdots+\frac{n+1}{3^{n}+1}\right)_{n=1}^{\infty}$ converges"
OK so we were shown some examples on how to prove such problems, so the first thing I did was to write it as a summation sequence: $1+\sum_{k=1}^{n}\frac{k+1}{3^{k}+1}$
Now I know that in order to prove that it is a Cauchy sequence I should show that for any $ (\epsilon>0) $ There exists an $N\in\mathbb{N}$ such that for any $n,m>N$ and $n>m$ the following occurs: $(|a_{n}-a_{m}|<\epsilon)$
This is where I'm stuck, I just can't seem to find the right N such that it works. Everything I've tried just doesn't seem to work.
Any pushes in the right direction are welcome! Thanks!