.......advanced calculus.... pove that:: :::Σ_(n=0) ^∞ {(((−1)^n )/2^(2n) ) ((( 2n)),(( n)) ) cos(nx)}=((cos((x/4)))/( (√(2cos((x/2))))))


Question and Answers Forum

All Questions Topic List
Differentiation Questions
Previous in All Question Next in All Question
Previous in Differentiation Next in Differentiation
Question Number 136581 by mnjuly1970 last updated on 23/Mar/21
$.......advanced calculus.... pove that:: :::Σ_(n=0) ^∞ {(((−1)^n )/2^(2n) ) ((( 2n)),(( n)) ) cos(nx)}=((cos((x/4)))/( (√(2cos((x/2))))))$
$. . . . . . . a d v a n c e d c a l c u l u s . . . .$ $p o v e t h a t ::$ $::: \underset{n = 0}{\sum^{\infty}} {\frac{{(- 1)}^{n}}{2^{2 n}} (\begin{matrix} 2 n \\ n \end{matrix}) c o s (n x)} = \frac{c o s (\frac{x}{4})}{\sqrt{2 c o s (\frac{x}{2})}}$
	Answered by Dwaipayan Shikari last updated on 23/Mar/21

	$\frac{1}{2} \underset{n = 0}{\sum^{\infty}} (\begin{matrix} 2 n \\ n \end{matrix}) {(- \frac{e^{i x}}{2^{2}})}^{n} + (\begin{matrix} 2 n \\ n \end{matrix}) {(- \frac{e^{- i x}}{4})}^{n}$ $= \frac{1}{2} . \frac{1}{\sqrt{1 + e^{i x}}} + \frac{1}{2 \sqrt{1 + e^{- i x}}} = \frac{e^{\frac{i x}{2}} + 1}{2 \sqrt{e^{i x} + 1}} = \frac{1 + c o s (\frac{x}{2}) + i s i n (\frac{x}{2})}{2 \sqrt{1 + c o s (x) + i s i n (x)}}$ $= \frac{2 c o s (\frac{x}{4}) (c o s \frac{x}{4} + i s i n (\frac{x}{4}))}{2 \sqrt{2 c o s \frac{x}{2}} \sqrt{c o s \frac{x}{2} + i s i n \frac{x}{2}}} = \frac{c o s (\frac{x}{4})}{\sqrt{2 c o s \frac{x}{2}}} . \frac{e^{i x / 4}}{e^{i x / 4}} = \frac{c o s (\frac{x}{4})}{\sqrt{2 c o s \frac{x}{2}}}$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum