How can I solve the sum of this series without differentiation?

[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?

 

[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

 

[hustler]

Perhaps the geometric series?

 

[jamjar]

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

 

[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?

 

[hustler]

hmmmmmmmmmmmm

 

[jamjar]

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

 

[Galileo]

I could differentiate perhaps?

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

 

[jamjar]

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

 

[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).