Looks like the starting series is
[tex]\displaystyle\sum_{n=0}^\infty\left((-1)^n+1\right)(n+1)x^n[/tex]
Note the degree of the first few terms in the series, for [tex]n=0,1,2[/tex], are [tex]x^0,x^1,x^2[/tex], and so on.
So if the series were to start at [tex]n=1[/tex], we would need to preserve that pattern, and we do that by replacing [tex]n[/tex] with [tex]n-1[/tex]:
[tex]\displaystyle\sum_{n=1}^\infty\left((-1)^{n-1}+1\right)(n-1+1)x^{n-1}[/tex]
Simplifying a bit, we end up with
[tex]\displaystyle\sum_{n=1}^\infty\left(1-(-1)^n\right)nx^{n-1}[/tex]
