Find the sum to infinity 1−(1/(2 ))cos θ+(1/4)cos 2θ−(1/8)cos 3θ+..........


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Question Number 54700 by peter frank last updated on 09/Feb/19

$F i nd the sum to infinity$ $1 - \frac{1}{2} \cos θ + \frac{1}{4} \cos 2 θ - \frac{1}{8} \cos 3 θ + . . . . . . . . . .$
Commented by maxmathsup by imad last updated on 09/Feb/19

$w e h a v e S = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{2^{n}} c o s (n θ) \Rightarrow S = R e (\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{2^{n}} e^{i n θ}) b u t$ $\sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{2^{n}} e^{i n θ} = \sum_{n = 0}^{\infty} {(- \frac{1}{2} e^{i θ})}^{n} (w e h a v e ∣ - \frac{1}{2} e^{i θ} ∣< 1)$ $= \frac{1}{1 + \frac{1}{2} e^{i θ}} = \frac{2}{2 + c o s θ + i s i n θ} = \frac{2 (2 + c o s θ - i s i n θ)}{{(2 + c o s θ)}^{2} + {s i n}^{2} θ} \Rightarrow$ $S = \frac{2 (2 + c o s θ)}{4 + 4 c o s θ + 1} = \frac{4 + 2 c o s θ}{5 + 4 c o s θ}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 09/Feb/19

	$p = 1 - \frac{c o s θ}{2} + \frac{c o s 2 θ}{4} - \frac{c o s 3 θ}{8} + . . .$ $q = - \frac{s i n θ}{2} + \frac{s i n 2 θ}{4} - \frac{s i n 3 θ}{8} + . . .$ $p + i q = 1 - \frac{1}{2} (c o s θ + i s i n θ) + \frac{1}{4} (c o s 2 θ + i s i n 2 θ) - \frac{1}{8} (c o s 3 θ + i s i n 3 θ) + . . .$ $p + i q = 1 - \frac{e^{i θ}}{2} + \frac{e^{i 2 θ}}{2^{2}} - \frac{e^{i 3 θ}}{2^{3}} + . . .$ $p + i q = 1 - a + a^{2} - a^{3} + . . .$ $p + i q = \frac{1}{1 + a} = \frac{1}{1 + \frac{e^{i θ}}{2}} = \frac{2}{2 + e^{i θ}} = \frac{2}{2 + c o s θ + i s i n θ}$ $p + i q = \frac{2 (2 + c o s θ - i s i n θ)}{{(2 + c o s θ)}^{2} + {(s i n θ)}^{2}}$ $p + i q = \frac{4 + 2 c o s θ}{4 + 4 c o s θ + 1} + \frac{- 2 i s i n θ}{5 + 4 c o s θ}$ $s o r e q u i r e d a n s i s = \frac{4 + 2 c o s θ}{5 + 4 c o s θ}$ $2$
	Commented by peter frank last updated on 09/Feb/19

	$t h a n k y o u s i r$
	Commented by peter frank last updated on 09/Feb/19

	$t h a n k y o u$
	Commented by peter frank last updated on 09/Feb/19

	$s o r r y s i r e l l a b o r a t e$ $s e c o n d l i n e$
	Commented by tanmay.chaudhury50@gmail.com last updated on 09/Feb/19

	$p = t h e s u m o f s e r i e s o f c o s θ$ $q = s u m o f s e r i e s o f s i n θ$ $t h e m e t h o d i s f i n d p + i q a n d c o n v e r t i t t o s e r i e s o f e^{i θ}$ $t h e n s e p a r a t e r e a l s n d i m a g i n a r y p a r t$ $r e a l p a r t \to p i m a g i n a r y p a r t \to q$
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