Sum of the series ∑ 2/(n²+15n+54) from n=0 to infinity

[aces9113]

Homework Statement

∞
∑ 2/(n^2+15n+54)
n = 0

Homework Equations

sum=a/1-r

The Attempt at a Solution

I've tried many different combinations trying to use the above equation but I'm kinda lost. i tried to do 2/70 divided by 1-(2/70) and various other combinations but i can't seem to get the right answer(although i don't know what it is yet

 

[sutupidmath]

Are you trying to determine whether the series converges, or finding the actual sum?

If the latter, I'm not totally sure whether this will work (since i haven't worked it out myself) but try to factor the denominator, then use partial fractions, and maybe the series will turn into a telescoping one...then you shouldn't have much trouble finding the sum.

 

[gomunkul51]

This is NOT a geometric series, so the sum: "sum=a/1-r" will not work.
This is surly a telescopic series. Try to write it as: 2/(n-r₁)(n-r₂)
r₁ and r₂ being the roots of the quadratic in the denominator.
Decompose it into its partial fraction of the form: 2/(n-r₁)(n-r₂) = A/(n-r₁) + B/(n-r₂).
Write the first several terms until you notice they start to cancel out...