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Question Number 80814 by mathmax by abdo last updated on 06/Feb/20
$${find}\:{nature}\:{of}\:{the}\:{serie}\:\sum_{{n}=\mathrm{1}} ^{\infty} \left(\mathrm{1}−{cos}\left(\frac{\pi}{{n}}\right)\right) \\ $$
Commented by mind is power last updated on 06/Feb/20
$${cos}\left(\frac{\pi}{{n}}\right)=\mathrm{1}−\frac{\pi^{\mathrm{2}} }{\mathrm{2}{n}^{\mathrm{2}} }+{o}\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\right) \\ $$$$\Rightarrow\mathrm{1}−{cos}\left(\frac{\pi}{{n}}\right)\sim\frac{\pi^{\mathrm{2}} }{\mathrm{2}{n}^{\mathrm{2}} }\:\:{cv}\:{Riemann} \\ $$
Commented by mathmax by abdo last updated on 06/Feb/20
$${thanks}\:{sir}. \\ $$