calcul-R-oo-n-o-oo-2n-2n-1-x-n-

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

c a l c u l : R = + o o

\underset{n = o}{\sum^{+ o o}} \frac{2 n}{(2 n + 1)!} x^{n}

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

\sinh x = \underset{n = 0}{\sum^{\infty}} \frac{x^{2 n + 1}}{(2 n + 1)!}

\frac{\sinh x}{x} = \underset{n = 0}{\sum^{\infty}} \frac{x^{2 n}}{(2 n + 1)!}

{(\frac{\sinh x}{x})}^{'} = \underset{n = 0}{\sum^{\infty}} \frac{2 {n x}^{2 n - 1}}{(2 n + 1)!}

x {(\frac{\sinh x}{x})}^{'} = \underset{n = 0}{\sum^{\infty}} \frac{2 {n x}^{2 n}}{(2 n + 1)!}

\cosh x - \frac{\sinh x}{x} = \underset{n = 0}{\sum^{\infty}} \frac{2 {n x}^{2 n}}{(2 n + 1)!}

r e p l a c e x w i t h \sqrt{x}

\Rightarrow \cosh \sqrt{x} - \frac{\sinh \sqrt{x}}{\sqrt{x}} = \underset{n = 0}{\sum^{\infty}} \frac{2 {n x}^{n}}{(2 n + 1)!}

Commented by Tawa11 last updated on 06/Jul/22

Great sir

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