prove-that-n-0-1-3n-e-3-2-3-e-cos-3-2-

Question Number 139457 by mnjuly1970 last updated on 27/Apr/21

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p r o v e t h a t ::

\underset{n = 0}{\sum^{\infty}} \frac{1}{(3 n)!} = \frac{e}{3} + \frac{2}{3 \sqrt{e}} c o s (\frac{\sqrt{3}}{2})

Answered by Dwaipayan Shikari last updated on 27/Apr/21

\underset{n = 0}{\sum^{\infty}} \frac{x^{3 n}}{(3 n)!} = \frac{e^{x} + e^{ω x} + e^{ω^{2} x}}{3} ω = e^{2 π i / 3}

= \frac{e^{x} + 2 e^{- \frac{x}{2}} c o s (\frac{\sqrt{3}}{2} x)}{3}

x = 1 :\to \underset{n = 0}{\sum^{\infty}} \frac{1}{(3 n)!} = \frac{e + 2 e^{- \frac{1}{2}} c o s (\frac{\sqrt{3}}{2})}{3}

Commented by Dwaipayan Shikari last updated on 27/Apr/21

F o r n = 4

i t i s \underset{n = 0}{\sum^{\infty}} \frac{x^{4 n}}{(4 n)!} = \frac{\underset{i = 1}{\sum^{R o o t s o f x^{4} = 1}} e^{a_{1} x}}{4} = \frac{e^{- i x} + e^{i x} + e^{x} + e^{- x}}{4}

= \frac{c o s h (x) + c o s (x)}{2}

F o r n = 2

\underset{n = 0}{\sum^{\infty}} \frac{x^{2 n}}{(2 n)!} = \frac{\underset{i = 1}{\sum^{R o o t s o f x^{2} = 1}} e^{a_{i} x}}{2} = \frac{e^{x} + e^{- x}}{2} = c o s h (x)

F o r n = 1

\underset{n = 0}{\sum^{\infty}} \frac{x^{n}}{n!} = \frac{\underset{i = 1}{\sum^{R o o t s o f x = 1}} e^{a_{i} x}}{1} = e^{x}

Commented by mnjuly1970 last updated on 27/Apr/21

v e r y n i c e \dots

Commented by Dwaipayan Shikari last updated on 27/Apr/21

B u t i s i t t r u e f o r

\underset{n = 0}{\sum^{\infty}} \frac{x^{a n}}{(a n)!} = \frac{\underset{i = 1}{\sum^{R o o t s o f x^{a} = 1}} e^{a_{i} x}}{a} ? ? A n y i d e a s i r ?

Commented by Dwaipayan Shikari last updated on 27/Apr/21

I f i g o f o r

\underset{n = 0}{\sum^{\infty}} \frac{x^{6 n}}{(6 n)!} = \frac{\underset{i = 1}{\sum^{R o o t s o f x^{6} = 1}} e^{a_{i} x}}{6} = \frac{e^{x} + e^{- x} + e^{\frac{1}{2} x} (e^{\frac{\sqrt{3}}{2} i x} + e^{- \frac{\sqrt{3}}{2} i x}) + e^{- \frac{x}{2}} (e^{\frac{\sqrt{3}}{2} i x} + e^{- \frac{\sqrt{3}}{2} i x})}{6}

= \frac{e^{x} + e^{- x} + 2 c o s (\frac{\sqrt{3}}{2} x) (e^{x / 2} + e^{- x / 2})}{6} = \frac{c o s h (x) + 2 c o s (\frac{\sqrt{3}}{2} x) c o s h (\frac{x}{2})}{3}

? ? ?