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Question Number 172311 by mathocean1 last updated on 25/Jun/22

Using Riemann′s sum, calculate:  lim b_n =(1/n)Σ_(k=0) ^(n−1) cos2(((kn)/n))

$${Using}\:{Riemann}'{s}\:{sum},\:{calculate}: \\ $$$${lim}\:{b}_{{n}} =\frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} {cos}\mathrm{2}\left(\frac{{kn}}{{n}}\right) \\ $$

Commented by JDamian last updated on 25/Jun/22

!!!  ((kn)/n)=k

$$!!!\:\:\frac{{k}\cancel{{n}}}{\cancel{{n}}}={k} \\ $$

Commented by thfchristopher last updated on 25/Jun/22

May I ask whether it can prove this:  cos^n x=(1/2^(n+1) )Σ_(k=0) ^((n−1)/2) C_k ^n cos (n−2k)x  where n is any odd number.

$$\mathrm{May}\:\mathrm{I}\:\mathrm{ask}\:\mathrm{whether}\:\mathrm{it}\:\mathrm{can}\:\mathrm{prove}\:\mathrm{this}: \\ $$$$\mathrm{cos}^{{n}} {x}=\frac{\mathrm{1}}{\mathrm{2}^{{n}+\mathrm{1}} }\underset{{k}=\mathrm{0}} {\overset{\left({n}−\mathrm{1}\right)/\mathrm{2}} {\sum}}{C}_{{k}} ^{{n}} \mathrm{cos}\:\left({n}−\mathrm{2}{k}\right){x} \\ $$$$\mathrm{where}\:{n}\:\mathrm{is}\:\mathrm{any}\:\mathrm{odd}\:\mathrm{number}. \\ $$

Commented by aleks041103 last updated on 25/Jun/22

See Q.170800 .

$${See}\:{Q}.\mathrm{170800}\:. \\ $$

Commented by thfchristopher last updated on 25/Jun/22

Thank you, sir.

$$\mathrm{Thank}\:\mathrm{you},\:\mathrm{sir}. \\ $$

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