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# Prove that the sum from 1 to ∞ of n / ((n+1)(n+2)(n+3)) = 1/4

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

1. tom says:

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…

• MathlessRick says:

For tom:
Notice that the numerator has degree 1 and denominator 3, so the difference is 2.
He wants to separate in two parts with a degree 2 in the denominator, so to keep the difference between the powers: 2 – 1 = 1 on the numerator

• MathlessRick says:

Ops i mean degree zero, because 2-2 =0
Sorry I kinda forgot how to subtract xD

• MathlessRick says:

I meant 2-2=0, degree 0

• Anonymous says:

i actually meant 2-2=0 so degree zero

• MathlessRick says:

Sorry, I meant degree zero
2-2=0

• Definitely not rick says:

Damn, it thought my internet wasn’t sending the messages xD
Well its 2-2=0 anyways