<s>[INST] Catalan's constant I am studying the integral $$\int_1^\infty\!\ \frac{\ln(x)}{1+x^{2}}$$ and ran across this representation of :
$$  \frac{1}{1+x^{2}}=\frac{1}{\frac{1}{x^{2}}(1-\frac{1}{x^{2}}+\frac{1}{x^{4}}-\frac{1}{x^{6}}+\frac{1}{x^{8}}...)}$$
where the denominator in the original integral has been rewritten to turn the integral into the following expression :
$$\int_1^\infty\!\frac{\ln(x)}{x^{2}}-\int_1^\infty\frac{\ln(x)}{x^{4}}+\int_1^\infty\frac{\ln(x)}{x^{6}} ...$$
which can then be evaluated by integrating by parts. I can do the integral but am stumped as to how the representation of $$\frac{1}{1+{x^{2}}}= \frac{1}{\frac{1}{x^{2}}(1-\frac{1}{x^{2}}+\frac{1}{x^{4}}-\frac{1}{x^{6}}+\frac{1}{x^{8}}...)}$$
was derived, and am assuming that the purpose was to get powers of $x$ into the denominator as opposed to having $\ln(x)$ multiplying the powers of $x$. Any help would be appreciated.
 [/INST] The correct representation is:
$$
\frac{1}{1+x^2}=\frac{1}{x^2}\frac{1}{1+x^{-2}}=\frac{1}{x^2}\sum_{k=0}^\infty\left(-\frac{1}{x^2}\right)^k,
$$
which you incorrectly put in the denominator of RHS.
</s>