Using n>=N ==> | A_n - L | < e

Most the proofs I read say to start with | a_n - 1 | < e and work backwards to find N but I have done so and I cannot eliminate then n term. I am stuck and any tips / help would be hugely appreciated.

I end up getting

1+2n - (-1)^n

-------------- < e

n^2 +2n+1