This problem has to do with Telescoping Sums Method.
Summation 2 / (n^2 + 4n + 3)
The method the professor taught us was to treat this like a Partial Fractions problem, break it up, find A and B and the result from what I get is
lim t-> infinity, summation ((1 / (n+1)) - (1 / (n+3)).
From here he told us to find the first few sequences by plugging in 1, 2, 3, 4, 5, 6 etc until we notice a pattern of terms cancelling out. All up to this part is cake... what i'm having trouble understanding is how to get the very last few terms where the terms will have t in it. How am I supposed to figure this out? Thanks