Basic Mathematics. Trouble with proof, powers and odd numbers.
$( -1)^n = ( -1 )^{2m+1} \color{red}= ( ( -1)^2 )^{m+1}$
Wrong. Note that $2(m+1)=2m+2$.
$(-1)^n = (-1)^{2k+1} \color{blue}= (-1)^{2k}(-1)$, for the blue equality, answer this question: how do you define $p^q$, where $p,q\in \Bbb Z$ and $q>0$?
You made an error in the 2nd equality. The relevant property of exponentiation here is: $$ a^{m+n}=a^m a^n $$