Question Number 31531 by abdo imad last updated on 09/Mar/18
$${let}\:{give}\:{f}\left({x}\right)=\frac{\mathrm{1}}{{x}}\:+\sum_{{n}=\mathrm{1}} ^{\infty} \:\left(\frac{\mathrm{1}}{{x}+{n}}\:+\frac{\mathrm{1}}{{x}−{n}}\right)\:{with}\:{x}\in{R}−{Z} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{f}\:{is}\:\mathrm{1}−{periodic} \\ $$$$\left.\mathrm{3}\right){prove}\:{that}\:{f}\left(\frac{{x}}{\mathrm{2}}\right)\:+{f}\left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)=\mathrm{2}{f}\left({x}\right). \\ $$