# why are these two power series the same

$$-\sum_{\color{red}{n=1}}^{\infty}nc_{n}x^{n}=-\sum_{\color{red}{n=0}}^{\infty}nc_{n}x^{n}$$

How come one starts at $1$ and the other starts at $0$ yet their equal? Do they both equal infinity?

• What is the value of the $0$-th term of the second sum? – Dario Nov 24 '14 at 22:47
• Same reason $1+2+3+\cdots=0+1+2+\cdots$ – JohnD Nov 24 '14 at 22:48

When $n=0$, $nc_nx^n$ is also zero. So the first sum just leaves that zero term out while the second one includes it. There is no change in the sum.