Find-the-sum-to-infinity-1-1-2-cos-1-4-cos-2-1-8-cos-3-

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 + \dots \dots \dots .

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} + \dots

q = - \frac{s i n θ}{2} + \frac{s i n 2 θ}{4} - \frac{s i n 3 θ}{8} + \dots

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) + \dots

p + i q = 1 - \frac{e^{i θ}}{2} + \frac{e^{i 2 θ}}{2^{2}} - \frac{e^{i 3 θ}}{2^{3}} + \dots

p + i q = 1 - a + a^{2} - a^{3} + \dots

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