# Sum of this eries: $\sum_{k=1}^{\infty}kp(1-p)^{k-1}$

$\sum_{k=1}^{\infty}kp(1-p)^{k-1}$

Can someone help me evaluate this sum? I couldn't even start, I have just written down the first couple of elements, but didn't help either.

Thanks!

• This may give you some ideas. Commented Apr 23, 2014 at 14:16

Can you compute the sum $$\sum_{k=1}^\infty (1-p)^k?$$ Now what happens if you differentiate term-by-term?
• You get something very close. $-\sum_{k=1}^\infty k(1-p)^{k-1}$. What do you have to do to get the original and how is that related to what the geometric series in my answer converges to? Commented Apr 25, 2014 at 14:51
• Lookup the theorem about differentiating term-by-term. Hint: Yes, you can differentiate the expression $1/p$ and get $-\sum_{k=1}^\infty k(1-p)^{k-1}=-1/p^2$... Based on what I just wrote what must the answer then be? Commented Apr 25, 2014 at 14:53
let the sum begin at k=0 and write instead of $kp(1-p)^k$ the expression $(k+1)p(1-P)^k$. Then expand the expression behind the sigma sign.