Question Number 172311 by mathocean1 last updated on 25/Jun/22
$${Using}\:{Riemann}'{s}\:{sum},\:{calculate}: \\ $$$${lim}\:{b}_{{n}} =\frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\mathrm{2}\left(\frac{{kn}}{{n}}\right) \\ $$
Commented by JDamian last updated on 25/Jun/22
$$!!!\:\:\frac{{k}\cancel{{n}}}{\cancel{{n}}}={k} \\ $$
Commented by thfchristopher last updated on 25/Jun/22
$$\mathrm{May}\:\mathrm{I}\:\mathrm{ask}\:\mathrm{whether}\:\mathrm{it}\:\mathrm{can}\:\mathrm{prove}\:\mathrm{this}: \\ $$$$\mathrm{cos}^{{n}} {x}=\frac{\mathrm{1}}{\mathrm{2}^{{n}+\mathrm{1}} }\underset{{k}=\mathrm{0}} {\overset{\left({n}−\mathrm{1}\right)/\mathrm{2}} {\sum}}{C}_{{k}} ^{{n}} \mathrm{cos}\:\left({n}−\mathrm{2}{k}\right){x} \\ $$$$\mathrm{where}\:{n}\:\mathrm{is}\:\mathrm{any}\:\mathrm{odd}\:\mathrm{number}. \\ $$
Commented by aleks041103 last updated on 25/Jun/22
$${See}\:{Q}.\mathrm{170800}\:. \\ $$
Commented by thfchristopher last updated on 25/Jun/22
$$\mathrm{Thank}\:\mathrm{you},\:\mathrm{sir}. \\ $$