The power series I have is derived from the sequence $$x+2x^2+x^3+2x^4+x^5+2x^6$$ And what I have is $$\sum^\infty_{n=0}x^{2n+1}+\sum^\infty_{n=1}2x^{2n+2}$$ seeing that the $n$ starts at different values we need to have them start at the same value. So I used $k=n-1$ then $n=k+1$ and when I reindexed I got
$$\sum^\infty_{k=0}x^{2k+3}+2x^{2k+2}$$ Am I correct or did I misunderstand what I am meant to do when reindexing?
You are making a mistake in 2nd line. The lower limit of the second sum should be n=0, and then also you don't have to reindex , $$ x+2x^2+x^3+2x^4+x^5+2x^6$$ So therefore $$\sum^\infty_{n=0}x^{2n+1}+\sum^\infty_{n=0}2x^{2n+2} =\sum^\infty_{n=0}( x^{2n+1} + 2x^{2n+2}) =\sum_{k=0}^\infty 2^{\frac {1}{2} [1+(-1)^n] }x^n $$