Σ_(n=1) ^∞ (n/(n!))cos(((πn)/5))


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Question Number 127793 by Dwaipayan Shikari last updated on 02/Jan/21

$\underset{n = 1}{\sum^{\infty}} \frac{n}{n!} c o s (\frac{π n}{5})$
Commented by Dwaipayan Shikari last updated on 03/Jan/21

$I h a v e f o u n d$ $e^{\frac{\sqrt{5} + 1}{4}} c o s (\frac{4 π + 5 \sqrt{2 (5 - \sqrt{5})}}{20}) = e^{\sqrt{φ}} c o s (\frac{4 π + 5 \sqrt{2 (5 - \sqrt{5})}}{20})$
	Answered by mnjuly1970 last updated on 02/Jan/21

	$\underset{n = 1}{\sum^{\infty}} \frac{1}{(n - 1)!} c o s (\frac{n π}{5})$ $= R e (\underset{n = 0}{\sum^{\infty}} \frac{e^{\frac{i (n - 1) π}{5}}}{n!}) = R e (e^{\frac{- i π}{5}} Σ \frac{e^{\frac{i n π}{5}}}{n!})$ $= c o s (\frac{π}{5}) R e [\underset{n = 0}{\sum^{\infty}} (\frac{e^{\frac{i n π}{5}}}{n!})]$ $= c o s (\frac{π}{5}) R e (e^{i \frac{π}{5}}) = {c o s}^{2} (\frac{π}{5})$ $= {(\frac{1 + \sqrt{5}}{4})}^{2} = \frac{6 + 2 \sqrt{5}}{16} = \frac{3 + \sqrt{5}}{8}$
	Answered by mathmax by abdo last updated on 02/Jan/21
	$S =Σ_(n=1) ^∞ (1/((n−1)!))cos(((nπ)/5)) ⇒S =Re(Σ_(n=1) ^∞ (1/((n−1)!))e^((inπ)/5) ) we have Σ_(n=1) ^∞ (1/((n−1)!)) (e^((iπ)/5) )^n =Σ_(n=0) ^∞ (1/(n!))(e^((iπ)/5) )^(n+1) =e^((iπ)/5) Σ_(n=0) ^∞ (((e^((iπ)/5) )^n )/(n!)) =e^((iπ)/5) ×e^e^((iπ)/5) =e^(((iπ)/5)+cos((π/5))+isin((π/5))) =e^(cos((π/5))+i((π/5) +sin((π/5))) =e^(cos((π/5))) ×{cos((π/5)+sin((π/5)))+isin((π/5) +sin((π/5)))} ⇒ S =e^(cos((π/5))) .cos((π/5)+sin((π/5))) with cos((π/5))=((1+(√5))/4)$
	$S = \sum_{n = 1}^{\infty} \frac{1}{(n - 1)!} \cos (\frac{n π}{5}) \Rightarrow S = Re (\sum_{n = 1}^{\infty} \frac{1}{(n - 1)!} e^{\frac{in π}{5}}) we have$ $\sum_{n = 1}^{\infty} \frac{1}{(n - 1)!} {(e^{\frac{i π}{5}})}^{n} = \sum_{n = 0}^{\infty} \frac{1}{n!} {(e^{\frac{i π}{5}})}^{n + 1} = e^{\frac{i π}{5}} \sum_{n = 0}^{\infty} \frac{{(e^{\frac{i π}{5}})}^{n}}{n!}$ $= e^{\frac{i π}{5}} \times e^{e^{\frac{i π}{5}}} = e^{\frac{i π}{5} + \cos (\frac{π}{5}) + isin (\frac{π}{5})} = e^{\cos (\frac{π}{5}) + i (\frac{π}{5} + \sin (\frac{π}{5})}$ $= e^{\cos (\frac{π}{5})} \times {\cos (\frac{π}{5} + \sin (\frac{π}{5})) + isin (\frac{π}{5} + \sin (\frac{π}{5}))} \Rightarrow$ $S = e^{\cos (\frac{π}{5})} . \cos (\frac{π}{5} + \sin (\frac{π}{5})) with \cos (\frac{π}{5}) = \frac{1 + \sqrt{5}}{4}$
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