Question Number 33125 by prof Abdo imad last updated on 10/Apr/18
$${let}\:{give}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{cos}\left({nx}\right){dx}}{\mathrm{1}−\mathrm{2}\lambda{cosx}\:+\lambda^{\mathrm{2}} } \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\lambda\:{u}_{{n}+\mathrm{2}} \:−\left(\mathrm{1}+\lambda^{\mathrm{2}} \right){u}_{{n}+\mathrm{1}} \:+\lambda\:{u}_{{n}} =\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\:{ptove}\:{that}\:\Sigma\:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{sum} \\ $$