n-o-oo-x-3n-2n-

Question Number 186689 by SANOGO last updated on 08/Feb/23

\underset{n = o}{\sum^{+ o o}} \frac{x^{3 n}}{(2 n)!} = ?

Commented by mr W last updated on 08/Feb/23

w h y n o t \infty i n s t e a d o f o o ?

w h y n o t 0 i n s t e a d o f o ?

Answered by mr W last updated on 08/Feb/23

e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \frac{x^{5}}{5!} + \dots (i)

e^{- x} = 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \frac{x^{4}}{4!} - \frac{x^{5}}{5!} + \dots (i i)

\frac{(i) + (i i)}{2} :

\frac{e^{x} + e^{- x}}{2} = 1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \dots = \underset{n = 0}{\sum^{\infty}} \frac{x^{2 n}}{(2 n)!}

r e p l a c e x w i t h x^{\frac{3}{2}}

\frac{e^{\frac{3 x}{2}} + e^{- \frac{3 x}{2}}}{2} = \underset{n = 0}{\sum^{\infty}} \frac{x^{3 n}}{(2 n)!}

o r

\cosh (\frac{3 x}{2}) = \underset{n = 0}{\sum^{\infty}} \frac{x^{3 n}}{(2 n)!} ✓

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