I have an exercise and the following sequences are given:
$b_n := (-1)^n$ $c_n:= (-1)^{n+1}\cdot n$
I have to prove the following for the power-series that belong to these sequences:
$P_c(x) + xP_c(x) = 1- P_b(x)$
I didn't make it very far though:
$\sum_{n=0}^\infty (-1)^{n+1} \cdot x^n+ x\cdot \sum_{n=0}^\infty (-1)^{n+1} x^n$ = $\sum_{n=0}^\infty (-1)^{n+1}\cdot x^n + (-1)^{n+1}\cdot x^{n+1}$
Helps and hints are appreciated!