Solving a series $n(1 + n + n^2 + n^3 + n^4 +…n^{n-1})$

Question

I'm trying to sum the following series?

$n(1 + n + n^2 + n^3 + n^4 +.......n^{n-1})$

Do you have any ideas?

Answer 1

This is called a geometric series.

$$n(1+n+n^2+\cdots n^{n-1})=n\frac{n^n-1}{n-1}$$

Why?

$$S=1+n+n^2+\cdots n^{n-1}$$

$$nS=n+n^2+n^3+\cdots n^{n}$$

$$S(1-n)=1-n^{n}$$

$$S=\frac{1-n^{n}}{1-n}$$