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Question Number 172989 by SANOGO last updated on 04/Jul/22

$$calcul:R=+oo$$$$\underset{n=o}{\sum ^{+oo}}\frac{2n}{(2n+1)!}{x}^n$$

Answered by mr W last updated on 04/Jul/22

$$\sinh x=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{x^{2n+1}}{(2n+1)!}$$$$\frac{\sinh x}{x}=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{x^{2n}}{(2n+1)!}$$$${\left(\frac{\sinh x}{x}\right)}^{\prime }=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{2{nx}^{2n-1}}{(2n+1)!}$$$$x{\left(\frac{\sinh x}{x}\right)}^{\prime }=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{2{nx}^{2n}}{(2n+1)!}$$$$\cosh x-\frac{\sinh x}{x}=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{2{nx}^{2n}}{(2n+1)!}$$$$replacexwith\sqrt{x}$$$$\Rightarrow \cosh \sqrt{x}-\frac{\sinh \sqrt{x}}{\sqrt{x}}=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{2{nx}^n}{(2n+1)!}$$

Commented by Tawa11 last updated on 06/Jul/22

$$\mathrm{Great}\mathrm{sir}$$

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