# I want to express the following power series 1 + x + x 2 </msup> + 3

I want to express the following power series
$1+x+{x}^{2}+3{x}^{3}+{x}^{4}+5{x}^{5}+{x}^{6}+7{x}^{7}+{x}^{8}+..$
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postojahob
Your series is the sum of
$1+{x}^{2}+{x}^{4}+{x}^{6}+\cdots =\frac{1}{1-{x}^{2}}$
with
$\begin{array}{rl}x+3{x}^{3}+5{x}^{5}+\cdots & =x\left(1+3{x}^{2}+5{x}^{4}+\cdots \right)\\ & =x{\left(x+{x}^{3}+{x}^{5}+\cdots \right)}^{\prime }\\ & =x{\left(\frac{x}{1-{x}^{2}}\right)}^{\prime }\\ & =\frac{{x}^{3}+x}{{\left({x}^{2}-1\right)}^{2}},\end{array}$
and therefore the sum of your series is
$\frac{1}{1-{x}^{2}}+\frac{{x}^{3}+x}{{\left({x}^{2}-1\right)}^{2}}=\frac{{x}^{3}-{x}^{2}+x+1}{{\left({x}^{2}-1\right)}^{2}}.$