Question

Write an equivalent series with the index of summation beginning at n = 1. ∞ (−1)n + 1(n + 1)xn n = 0

Answer 1

Looks like the starting series is

$\displaystyle\sum_{n=0}^\infty\left((-1)^n+1\right)(n+1)x^n$

Note the degree of the first few terms in the series, for $n=0,1,2$, are $x^0,x^1,x^2$, and so on.

So if the series were to start at $n=1$, we would need to preserve that pattern, and we do that by replacing $n$ with $n-1$:

$\displaystyle\sum_{n=1}^\infty\left((-1)^{n-1}+1\right)(n-1+1)x^{n-1}$

Simplifying a bit, we end up with

$\displaystyle\sum_{n=1}^\infty\left(1-(-1)^n\right)nx^{n-1}$