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

Question

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

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!

Answer 1

Can you compute the sum $$\sum_{k=1}^\infty (1-p)^k?$$ Now what happens if you differentiate term-by-term?

Answer 2

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.

greetings,

calculus.