# Thread: [SOLVED] telescoping sum with a partial fraction

1. ## [SOLVED] telescoping sum with a partial fraction

the sum from 2 to infinity of 2/(n^2 - 1)

so first i factor the denominator

2/((n + 1)(n - 1))

this can expressed with partial fractions

2/((n + 1)(n - 1)) = A/(n + 1) + B/(n - 1)

which can be rearanged to

2 = A(n + 1)(n - 1) + B(n + 1)(n - 1)

now I'm stuck. I have one equation and 2 variables. I know there is something I'm forgetting how to do this.

2. Originally Posted by atompunk
the sum from 2 to infinity of 2/(n^2 - 1)

so first i factor the denominator

2/((n + 1)(n - 1))

this can expressed with partial fractions

2/((n + 1)(n - 1)) = A/(n + 1) + B/(n - 1)

which can be rearanged to

2 = A(n + 1)(n - 1) + B(n + 1)(n - 1)

now I'm stuck. I have one equation and 2 variables. I know there is something I'm forgetting how to do this.
It should rearrange to $\displaystyle 2=A(n-1)+B(n+1)\implies 2=(A+B)n+B-A$.

Comparing coefficients, we get $\displaystyle A+B=0$ and $\displaystyle B-A=2$.

Can you take it from here and find A and B?

3. $\displaystyle 2=(n+1)-(n-1).$

4. thanks