calculate Σ_(n=1) ^∞ ((cos(n(π/3)))/n)


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Question Number 67232 by prof Abdo imad last updated on 24/Aug/19

$c a l c u l a t e \sum_{n = 1}^{\infty} \frac{c o s (n \frac{π}{3})}{n}$
Commented by mathmax by abdo last updated on 25/Aug/19

$f i r s t l e t d e t e r m i n e \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n} = R e (\sum_{n = 1}^{\infty} \frac{e^{i n x}}{n})$ $l e t s (x) = \sum_{n = 1}^{\infty} \frac{e^{i n x}}{n} \Rightarrow s^{^{'}} (x) = i \sum_{n = 1}^{\infty} e^{i n x} = {i e}^{i x} \sum_{n = 1}^{\infty} e^{i (n - 1) x}$ $= {i e}^{i x} \sum_{n = 0}^{\infty} e^{i n x} = \frac{{i e}^{i x}}{1 - e^{i x}} \Rightarrow s (x) = - l n (1 - e^{i x}) + c$ $x = π \Rightarrow \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{n} = - l n (2) = - l n (2) + c \Rightarrow c = 0$ $s (x) = - l n (1 - e^{i x})$ $l n (1 - e^{i x}) = l n (1 - c o s x - i s i n x) = l n (2 {s i n}^{2} (\frac{x}{2}) - 2 i s i n (\frac{x}{2}) c o s (\frac{x}{2}))$ $= l n (- 2 i s i n (\frac{x}{2}) e^{i \frac{x}{2}}) = l n (- 2 i) + l n (s i n (\frac{x}{2})) + i \frac{x}{2}$ $= l n (2) + l n (- i) + l n (s i n (\frac{x}{2})) + \frac{i x}{2}$ $= l n (2 s i n (\frac{x}{2})) - \frac{i π}{2} + \frac{i x}{2} \Rightarrow \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n} = - l n (2 s i n (\frac{x}{2}))$ $x = \frac{π}{3} \Rightarrow \sum_{n = 1}^{\infty} \frac{c o s (\frac{n π}{3})}{n} = - l n (2 s i n (\frac{π}{6})) = - l n (2 \times \frac{1}{2}) = 0 s o$ $w e h a v e \frac{c o s (\frac{π}{3})}{1} + \frac{c o s (\frac{2 π}{3})}{2} + \frac{c o s (\frac{3 π}{3})}{3} + . . . . = 0$ $a l s o w e c a n e x t r a c t t h a t \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} = \frac{π - x}{2} \Rightarrow$ $\sum_{n = 1}^{\infty} \frac{s i n (\frac{n π}{3})}{n} = \frac{π - \frac{π}{3}}{2} = \frac{π}{2} - \frac{π}{6} = \frac{2 π}{6} = \frac{π}{3}$
	Answered by Smail last updated on 25/Aug/19

	$\underset{n = 1}{\sum^{\infty}} \frac{e^{i n x}}{n} = - l n (1 - e^{i x})$ $= - l n (1 - c o s (x) - i s i n (x))$ $= - l n (2 {s i n}^{2} (x / 2) - 2 i s i n (x / 2) c o s (x / 2))$ $= - l n (2) - l n ∣ s i n (x / 2) ∣ - l n (s i n (x / 2) - i c o s (x / 2))$ $= - l n (2) - l n ∣ s i n (x / 2) ∣ - l n (c o s (\frac{π - x}{2}) - i s i n (\frac{π - x}{2}))$ $= - l n (2) - l n ∣ s i n (x / 2) ∣ + \frac{π - x}{2} i$
	Commented by Smail last updated on 25/Aug/19

	$F o r x = \frac{π}{3}$ $\underset{n = 1}{\sum^{\infty}} \frac{e^{i n \frac{π}{3}}}{n} = - l n (2) - l n ∣ s i n (π / 6) ∣ + i \frac{π - π / 3}{2}$ $S o, \underset{n = 1}{\sum^{\infty}} \frac{c o s (n \frac{π}{3})}{n} = R e (\underset{n = 1}{\sum^{\infty}} \frac{e^{i n \frac{π}{3}}}{n})$ $\underset{n = 1}{\sum^{\infty}} \frac{c o s (\frac{n π}{3})}{n} = - l n (2) - l n ∣ s i n (π / 6) ∣$
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