Converging a Series: Finding the Actual Value Using Integral Approximation

[mentil]

Homework Statement

v=sum of i*(i+1)/((1+y)^i) from i =1 to i=infinity

for y > 0

Homework Equations

The Attempt at a Solution

This isn't a homework question per se (and I'm not that interested in the actual number solution), but how would one go about solving v? Are there any general tricks? I tried searching online, but all I could find are ways to prove that it converges -- not that how to find the actual converging value. It easy to see that it converges, just having a little trouble finding out to what value it does.

 

[chiro]

Homework Statement

v=sum of i*(i+1)/((1+y)^i) from i =1 to i=infinity

for y > 0

Homework Equations

The Attempt at a Solution

This isn't a homework question per se (and I'm not that interested in the actual number solution), but how would one go about solving v? Are there any general tricks? I tried searching online, but all I could find are ways to prove that it converges -- not that how to find the actual converging value. It easy to see that it converges, just having a little trouble finding out to what value it does.

I don't know if this helps you but one trick that mathematicians use for these kind of problems is to use an integral approximation.

This can work depending on how big or small the error term is.

In a nutshell it works by simply finding the integral of the specified function, but the problem is that if the error is too large then its basically useless for the application you have in mind (ie finding a good approximation).

The Euler-Maclaurin Integration Formulas might be what you need:

http://mathworld.wolfram.com/Euler-MaclaurinIntegrationFormulas.html

Good luck!