Prove that the following sum converges and has the given value.

So, first we need to use partial fractions to decompose the terms in the sum,

This gives us the equations,

Therefore we have,

Using partial fraction decomposition again, we have and in the above equation, so we have

where . Therefore, by the theorem on telescoping sums (Theorem 10.7 of Apostol) we have

I always thought partial fraction decomposition required the denominator to be irreducible yet you have reducible quadratics; i.e. shouldn’t the numerators be in the AX+B format? Hopefully learning linear algebra down the road will clear this up…