Question

Sum the series
${(x+\frac{1}{x})}^2+{(x^2+\frac{1}{x^2})}^2+{(x^3+\frac{1}{x^3})}^2................+{(x^n+\frac{1}{x^n})}^2$

Answer 1

Open the brackets and split into three series
$(x^2+x^4+x^6+...)+(\frac{1}{x^2}+\frac{1}{x^4}+\frac{1}{x^6}+...)+(2+2+2+...)$
$=x^2\left(\frac{x^{2n}-1}{x^2-1}\right)+\frac{1}{x^2}\frac{(\frac{1}{x^{2n}}-1)}{\frac{1}{x^2}-1}+2n$
$=\frac{x^2(x^{2n}-1)}{x^2-1}+\frac{x^{2n}-1}{x^{2n}.(x^2-1)}+2n$
$=\frac{x^{2n}-1}{x^2-1}[x^2+\frac{1}{x^{2n}}]+2n$
$=\frac{x^{2n}-1}{x^2-1}\left[\frac{x^{2n+2}+1}{x^{2n}}\right]+2n$