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Question Number 133830 by metamorfose last updated on 24/Feb/21

$$find\underset{x\to 0}{\lim }\frac{1-cos\left(x\right){cos}^2\left(2x\right){cos}^3\left(3x\right)...{cos}^n\left(nx\right)}{x^2}=...?$$

Answered by EDWIN88 last updated on 24/Feb/21

$$\underset{x\to 0}{\lim }\frac{1-(1-\frac{1}{2}{\mathrm{x}}^2)(1-\frac{8}{2}{\mathrm{x}}^2)(1-\frac{27}{2}{\mathrm{x}}^2)...(1-\frac{{\mathrm{n}}^3}{2}{\mathrm{x}}^2)}{{\mathrm{x}}^2}=$$$$\underset{x\to 0}{\lim }\frac{1-(1-\frac{1}{2}{\mathrm{x}}^2(1^3+2^3+3^3+...+{\mathrm{n}}^3))}{{\mathrm{x}}^2}=$$$$\frac{1}{2}(1^3+2^3+3^3+...+{\mathrm{n}}^3)=\frac{1}{2}{\left[\frac{\mathrm{n}(\mathrm{n}+1)}{2}\right]}^2$$$$$$

Answered by Dwaipayan Shikari last updated on 24/Feb/21

$$\underset{x\to 0}{\lim }\frac{1-cos\left(x\right){cos}^2\left(2x\right){cos}^3\left(3x\right)...{cos}^n\left(nx\right)}{x^2}=y$$$$\underset{n=1}{\sum ^n}nlog\left(cos\left(nx\right)\right)=log(1-x^{2}y)\underset{x\to 0}{\lim }log(1+x)=x$$$$\underset{n=1}{\sum ^n}nlog(1-\frac{n^2{x}^2}{2})=log(1-x^{2}y)\underset{x\to 0}{\lim }cosx=1-\frac{x^2}{2}$$$$-\underset{n=1}{\sum ^n}\frac{n^3{x}^2}{2}=-x^{2}y\Rightarrow \frac{1}{2}\underset{n=1}{\sum ^n}{n}^3=y$$$$y=\frac{n^2{(n+1)}^2}{8}$$

Answered by bramlexs22 last updated on 24/Feb/21

$$\underset{x\to 0}{\lim }\frac{1-\cos \mathrm{x}\cos {}^{2}2\mathrm{x}}{{\mathrm{x}}^2}=$$$$\underset{x\to 0}{\lim }\frac{1-\cos \mathrm{x}{(1-2\sin {}^2\mathrm{x})}^2}{{\mathrm{x}}^2}=$$$$\underset{x\to 0}{\lim }\frac{1-(1-\frac{1}{2}{\mathrm{x}}^2)(1-{4\mathrm{x}}^2)}{{\mathrm{x}}^2}=$$$$\underset{x\to 0}{\lim }\frac{1-(1-\frac{9}{2}{\mathrm{x}}^2+\frac{1}{2}{\mathrm{x}}^4)}{{\mathrm{x}}^2}=\frac{9}{2}$$$$$$

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