1)find Σ_(n=1) ^∞ ((cos(nx))/n) and Σ_(n=1) ^∞ ((sin(nx))/n) 2) calculate Σ_(n=1) ^∞ (1/n)cos(((2nπ)/3)) and Σ


Question and Answers Forum

All Questions Topic List
Relation and Functions Questions
Previous in All Question Next in All Question
Previous in Relation and Functions Next in Relation and Functions
Question Number 45968 by maxmathsup by imad last updated on 19/Oct/18

$1) f i n d \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n} a n d \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n}$ $2) c a l c u l a t e \sum_{n = 1}^{\infty} \frac{1}{n} c o s (\frac{2 n π}{3}) a n d \sum_{n = 1}^{\infty} \frac{1}{n} s i n (\frac{2 n π}{3})$
Commented by maxmathsup by imad last updated on 23/Oct/18

$1) l e t f (x) = \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n} a n d g (x) = \sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} w e h a v e$ $f (x) + i g (x) = \sum_{n = 1}^{\infty} \frac{e^{i n x}}{n} = \sum_{n = 1}^{\infty} \frac{{(e^{i x})}^{n}}{n} = - l n (1 - e^{i x})$ $= - l n (1 - c o s x - i s i n x) = - l n (2 {s i n}^{2} (\frac{x}{2}) - 2 i s i n (\frac{x}{2}) c o s (\frac{x}{2}))$ $= - l n (- i s i n (\frac{x}{2}) (e^{i \frac{x}{2}})) = - l n (- i) + l n (s i n (\frac{x}{2})) - i \frac{x}{2}$ $= - l n (e^{- i \frac{π}{2}}) - i \frac{x}{2} + l n (s i n (\frac{x}{2})) = \frac{i π}{2} - i \frac{x}{2} + l n (s i n (\frac{x}{2}))$ $= i \frac{π - x}{2} + l n (s i n (\frac{x}{2})) \Rightarrow \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n} = l n (s i n (\frac{x}{2})) a n d$ $\sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} = \frac{π - x}{2} .$
Commented by maxmathsup by imad last updated on 23/Oct/18

$2) w e h a v e p r o v e d t h a t \sum_{n = 1}^{\infty} \frac{c o s (n x)}{n} = l n (s i n (\frac{x}{2}) f o r x = \frac{2 π}{3} \Rightarrow$ $\sum_{n = 1}^{\infty} \frac{c o s (\frac{2 n π}{3})}{n} = l n (s i n (\frac{π}{3})) = l n (\frac{\sqrt{3}}{2}) = \frac{1}{2} l n (3) - l n (2) a l s o w e h a v e$ $\sum_{n = 1}^{\infty} \frac{s i n (n x)}{n} = \frac{π - x}{2} \Rightarrow \sum_{n = 1}^{\infty} \frac{s i n (\frac{2 n π}{3})}{n} = \frac{π - \frac{2 π}{3}}{2} = \frac{π}{2} - \frac{π}{3} = \frac{π}{6} .$
	Answered by Smail last updated on 22/Oct/18

	$p (x) = \underset{n = 1}{\sum^{\infty}} \frac{e^{i n x}}{n}$ $l n (1 - x) = - \underset{n = 1}{\sum^{\infty}} \frac{x^{n}}{n} w i t h ∣ x ∣< 1$ $S o p (x) = - (- \underset{n = 1}{\sum^{\infty}} \frac{{(e^{i x})}^{n}}{n}) = - l n (1 - e^{i x})$ $p (x) = a + i b = - l n (1 - e^{i x})$ $\underset{n = 1}{\sum^{\infty}} \frac{e^{i n x}}{n} = \underset{n = 1}{\sum^{\infty}} \frac{c o s (n x)}{n} + i \underset{n = 1}{\sum^{\infty}} \frac{s i n (n x)}{n}$ $e^{a + i b} = \frac{1}{1 - c o s (x) - i s i n (x)}$ $e^{a} c o s (b) + {i e}^{a} s i n (b) = \frac{1 - c o s (x) + i s i n (x)}{2 (1 - c o s (x))}$ $e^{a} c o s (b) = \frac{1}{2} a n d e^{a} s i n (b) = \frac{s i n (x)}{2 (1 - c o s (x))}$ $t a n (b) = \frac{s i n (x)}{1 - c o s (x)}$ $b = {t a n}^{- 1} (\frac{s i n (x)}{1 - c o s (x)})$ $w h i c h i s b = \underset{n = 1}{\sum^{\infty}} \frac{s i n (n x)}{n} = {t a n}^{- 1} (\frac{s i n x}{1 - c o s x})$ $a n d e^{a} c o s (b) = \frac{1}{2}$ $e^{a} = \frac{1}{2 c o s (b)} = \frac{1}{2} \sqrt{1 + {t a n}^{2} (b)}$ $e^{a} = \frac{1}{2} \sqrt{1 + \frac{{s i n}^{2} x}{{(1 - c o s x)}^{2}}} = \frac{1}{2} \sqrt{\frac{2 (1 - c o s (x))}{{(1 - c o s x)}^{2}}}$ $= \frac{\sqrt{2}}{2} \sqrt{\frac{1}{1 - c o s x}} S o a = l n (\frac{1}{\sqrt{2}}) + l n (\frac{1}{\sqrt{1 - c o s x}})$ $a = \underset{n = 1}{\sum^{\infty}} \frac{c o s (n x)}{n} = - \frac{1}{2} (l n (2) + l n (1 - c o s x))$
	Commented by Smail last updated on 22/Oct/18

	$a (x) = - \frac{1}{2} (l n (2) + l n (1 - c o s x))$ $a (\frac{2 π}{3}) = \underset{n = 1}{\sum^{\infty}} \frac{c o s (2 n π / 3)}{n} = - \frac{1}{2} (l n (2) + l n (1 - c o s (2 π / 3)))$ $= - \frac{1}{2} (l n (2) + l n (1 + \frac{1}{2})) = - \frac{l n 3}{2}$ $b (x) = {t a n}^{- 1} (\frac{s i n x}{1 - c o s x})$ $b (2 π / 3) = {t a n}^{- 1} (\frac{s i n (2 π / 3)}{1 - c o s (2 π / 3)})$ $= {t a n}^{- 1} (\frac{\sqrt{3}}{3}) = \frac{π}{6} + k π$ $s o \underset{n = 1}{\sum^{\infty}} \frac{s i n (2 n π / 3)}{n} = \frac{π}{6}$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum