Help with an infinite series sum

I understand how to solve for convergence, but how would you solve for the actual sum of this problem:
( sigma n=0 to infinity ) 3/(n^2+17n+70)

Please help! I am really having trouble understanding how to solve these types of problems.

Factor
3/(n+10)(n+7)

Partial fraction decomposition
A/(n+10) + B/(n+7)
An + A7 + Bn + B10 =3
A=-B
B=1
A =-1

Therefore
1/(n+7) - 1/(n+10)

The first terms are
(1/7 - 1/10) + (1/8 - 1/11) + (1/9 - 1/12) + (1/10 - 1/13) + ...

Notice the second half of each term cancels with the first half of the term three terms later

We are left with the first half of the first three terms and the second half of the last three terms. But there are no "last three" - the second half of each term approaches zero as n approaches infinity

So we are really only left with the first half of the first three terms
1/7 + 1/8 + 1/9

( sigma n=0 to infinity ) 3/(n^2+17n+70)
= ( sigma n=0 to infinity ) [ 1/ (n + 7) -- 1/ (n + 10) ]
= 1/7 -- 1/10 + 1/8 -- 1/11 + 1/9 -- 1/12 + ...
= 1/7 + 1/8 + 1/9
= (72 + 63 + 56) / 7*8*9
= 191/504