calculate : Σ_(n=1) ^∞ (( (−1)^( n−1) )/n) cos ((( nπ)/3) ) =?


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Question Number 190623 by mnjuly1970 last updated on 07/Apr/23

$calculate :$ $\underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n - 1}}{n} \cos (\frac{n π}{3}) = ?$
	Answered by mahdipoor last updated on 07/Apr/23

	$g e t f (x) = x^{2} i n [- 3, 3]$ $a_{0} = \frac{1}{3} \int_{- 3}^{3} x^{2} . d x = 18$ $a_{n} = \frac{1}{3} \int_{- 3}^{3} x^{2} . c o s (\frac{n π x}{3}) =$ $= \frac{1}{3} {[\frac{x^{2} s i n (\frac{n π x}{3})}{\frac{n π}{3}} + \frac{2 x c o s (\frac{n π x}{3})}{\frac{n π}{3}} - \frac{2 s i n (\frac{n π x}{3})}{\frac{n π}{3}}]}_{- 3}^{3}$ $= \frac{2}{n π} [9 s i n (n π) + 6 c o s (n π) - 2 s i n (n π)] = \frac{12}{n π} {(- 1)}^{n}$ $b_{n} = \frac{1}{3} \int_{- 3}^{3} x^{2} . s i n (\frac{n π x}{3}) = 0$ $\Rightarrow\Rightarrow$ $x^{2} = \frac{18}{2} + \underset{n = 1}{\sum^{\infty}} (\frac{12 {(- 1)}^{n}}{n π} c o s (\frac{n π x}{3})) \Rightarrow x = 1 \Rightarrow$ $1 - \frac{18}{2} = - \frac{12}{π} \underset{n = 1}{\sum^{\infty}} (\frac{{(- 1)}^{n - 1}}{n} c o s (\frac{n π}{3})) \Rightarrow$ $\underset{n = 1}{\sum^{\infty}} (\frac{{(- 1)}^{n - 1}}{n} c o s (\frac{n π}{3})) = \frac{2 π}{3}$
	Commented by mnjuly1970 last updated on 07/Apr/23

	$a n s w e r : \frac{l n (3)}{2}$
	Answered by Peace last updated on 08/Apr/23

	$= R e \sum_{n ⩾ 1} \frac{{(- 1)}^{n - 1} {(e^{i \frac{π}{3}})}^{n}}{n}$ $= R e l n (1 + e^{i \frac{π}{3}})$ $= l n (∣ 1 + e^{i \frac{π}{3}} ∣) = R e (∣ e^{i \frac{π}{6}} ∣∣ 2 c o s (\frac{π}{6}) ∣)$ $= l n (\sqrt{3}) = \frac{l n (3)}{2}$
	Commented by mnjuly1970 last updated on 08/Apr/23

	$s o n i c e s i r t h x a l o t = .$
	Commented by mehdee42 last updated on 08/Apr/23

	$v e r y g o o d$
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