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Question Number 47070 by Meritguide1234 last updated on 04/Nov/18

Commented by maxmathsup by imad last updated on 04/Nov/18
$let S(x)=Σ_(n=0) ^∞ ((cos(nx))/(n!)) ⇒S(x) =Re(Σ_(n=0) ^∞ (e^(inx) /(n!))) but Σ_(n=0) ^∞ (e^(inx) /(n!)) =e^(ix) =Σ_(n=0) ^∞ (((e^(ix) )^n )/(n!)) =e^e^(ix) =e^(cosx +isinx) =e^(cosx) {cos(sinx)+isin(sinx)} ⇒ S(x) =e^(cosx) .cos(sinx) and Σ_(n=0) ^∞ ((cos(((πn)/6)))/(n!)) = S((π/6)) =e^((√3)/2) cos((1/2)) .$
$l e t S (x) = \sum_{n = 0}^{\infty} \frac{c o s (n x)}{n!} \Rightarrow S (x) = R e (\sum_{n = 0}^{\infty} \frac{e^{i n x}}{n!}) b u t$ $\sum_{n = 0}^{\infty} \frac{e^{i n x}}{n!} = e^{i x} = \sum_{n = 0}^{\infty} \frac{{(e^{i x})}^{n}}{n!} = e^{e^{i x}} = e^{c o s x + i s i n x} = e^{c o s x} {c o s (s i n x) + i s i n (s i n x)}$ $\Rightarrow S (x) = e^{c o s x} . c o s (s i n x) a n d \sum_{n = 0}^{\infty} \frac{c o s (\frac{π n}{6})}{n!} = S (\frac{π}{6}) = e^{\frac{\sqrt{3}}{2}} c o s (\frac{1}{2}) .$
Commented by Meritguide1234 last updated on 04/Nov/18

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