Series sum

1. Oct 19, 2005

jamjar

Hi,
I've come across this series and I'm not sure in which direction I should be looking to get an equation for the sum. I've tried some simple methods but have come up blank.
$$\sum\limits_{n = 0}^{n - 1} {nr^n }$$
Can anyone give me a nudge in the right direction?

2. Oct 19, 2005

Galileo

Does it look like another series you know? Can you find some way to relate the two?

BTW: Your index is n, but you are summing to n-1. So n is doing double duty. I suppose you mean:
$$\sum_{k=0}^{n-1}kr^k$$

3. Oct 19, 2005

hustler

Perhaps the geometric series?

Last edited: Oct 19, 2005
4. Oct 19, 2005

jamjar

I can't see how to relate the two.
The extra multiplication by k is making it difficult.

5. Oct 19, 2005

Galileo

Okay, here's where my ignorance about the contents of a pre-calculus class may come into play, but...the terms in the geometric series have $r^k$. Is there anything, some operation, you can do to each of the $r^k$ terms to make it look like more or less $kr^k$?

6. Oct 19, 2005

hustler

hmmmmmmmmmmmm

7. Oct 19, 2005

jamjar

I could differentiate perhaps?
I'm not sure what operations I can use within the summation.

8. Oct 19, 2005

Galileo

That's a good idea!
What'd you get if you differentiate a geometric sum?

9. Oct 19, 2005

jamjar

Well, I worked it out from there.
I just wasn't expecting to get any differentiation in pre-calc.
Thanks for the help

10. Oct 19, 2005

shmoe

You can do it without differentiation if you like. Write it as a double sum and swap order of summation. You could also think of this as writing it as a sum of geometric series (all of different lengths).