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Question Number 127486 by shaker last updated on 30/Dec/20

Answered by Dwaipayan Shikari last updated on 30/Dec/20

$$\underset{x\to 0}{\lim }\frac{\underset{k=1}{\prod ^n}cos\left(a_{k}x\right)-1}{x^2}=y$$$$\underset{x\to 0}{\lim }\underset{k=1}{\prod ^n}cos\left(a_{k}x\right)=x^{2}y+1\Rightarrow \underset{x\to 0}{\lim }\underset{k=1}{\sum ^n}log\left({cosa}_{k}x\right)=log(x^{2}y+1)$$$$=\underset{k=1}{\sum ^n}log(1-\frac{a_k^2{x}^2}{2})=x^{2}yAs\underset{x\to 0}{\lim }log(1+x)=xcosx=1-\frac{x^2}{2}$$$$\Rightarrow -\underset{k=1}{\sum ^n}\frac{a_k^2{x}^2}{2}=x^{2}y$$$$y=-\frac{1}{2}\underset{k=1}{\sum ^n}{a}_k^2$$

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