Question

How does summation work?

Summing from k=0 to n, in this function: p(1-p)^(n-1)/n?

Answer 1

I will take it you mean what's below. Since our summation variable does not affect our expression each sum, we then have:
\[
\sum_{k=0}^n \frac{p(1-p)^{n-1}}{n}=\underbrace{\frac{p(1-p)^{n-1}}{n}+\frac{p(1-p)^{n-1}}{n}+\cdots+\frac{p(1-p)^{n-1}}{n}}_{n+1\text{ times}}
\]Which equals:
\[
\sum_{k=0}^n \frac{p(1-p)^{n-1}}{n}=\frac{(n+1)p(1-p)^{n-1}}{n}
\]

Answer 2

Thank you, you god.