calculate-n-1-cos-n-pi-3-n-

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} + \dots . = 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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