Anybody know a proof of $prod_{n=1}^inftycos(x/2^n)=sin x/x$.











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This is actually an exercise from Apostol's Mathematical Analysis. Ch. 8 Ex 42. which asks to find all real values $x$ for which $prod_{n=1}^infty cosleft(largefrac{x}{2^n}right)$ converges. I've shown that the product converges for all $x$. The problem then asks to find what values the product converges to. By playing around with Wolfram Alpha, I found that
$$largeprod_{n=1}^inftycosleft(frac{x}{2^n}right)=frac{sin (x)}{x}.$$



I can't figure out how to prove this.










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  • 2




    It has the correct set of zeroes, hence the quotient is an entire analytic function
    – Hagen von Eitzen
    Mar 18 '15 at 17:49















up vote
12
down vote

favorite
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This is actually an exercise from Apostol's Mathematical Analysis. Ch. 8 Ex 42. which asks to find all real values $x$ for which $prod_{n=1}^infty cosleft(largefrac{x}{2^n}right)$ converges. I've shown that the product converges for all $x$. The problem then asks to find what values the product converges to. By playing around with Wolfram Alpha, I found that
$$largeprod_{n=1}^inftycosleft(frac{x}{2^n}right)=frac{sin (x)}{x}.$$



I can't figure out how to prove this.










share|cite|improve this question




















  • 2




    It has the correct set of zeroes, hence the quotient is an entire analytic function
    – Hagen von Eitzen
    Mar 18 '15 at 17:49













up vote
12
down vote

favorite
3









up vote
12
down vote

favorite
3






3





This is actually an exercise from Apostol's Mathematical Analysis. Ch. 8 Ex 42. which asks to find all real values $x$ for which $prod_{n=1}^infty cosleft(largefrac{x}{2^n}right)$ converges. I've shown that the product converges for all $x$. The problem then asks to find what values the product converges to. By playing around with Wolfram Alpha, I found that
$$largeprod_{n=1}^inftycosleft(frac{x}{2^n}right)=frac{sin (x)}{x}.$$



I can't figure out how to prove this.










share|cite|improve this question















This is actually an exercise from Apostol's Mathematical Analysis. Ch. 8 Ex 42. which asks to find all real values $x$ for which $prod_{n=1}^infty cosleft(largefrac{x}{2^n}right)$ converges. I've shown that the product converges for all $x$. The problem then asks to find what values the product converges to. By playing around with Wolfram Alpha, I found that
$$largeprod_{n=1}^inftycosleft(frac{x}{2^n}right)=frac{sin (x)}{x}.$$



I can't figure out how to prove this.







real-analysis infinite-product






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edited Dec 10 at 11:11









Gaby Alfonso

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670315










asked Mar 18 '15 at 17:46









Tim Raczkowski

17.3k21242




17.3k21242








  • 2




    It has the correct set of zeroes, hence the quotient is an entire analytic function
    – Hagen von Eitzen
    Mar 18 '15 at 17:49














  • 2




    It has the correct set of zeroes, hence the quotient is an entire analytic function
    – Hagen von Eitzen
    Mar 18 '15 at 17:49








2




2




It has the correct set of zeroes, hence the quotient is an entire analytic function
– Hagen von Eitzen
Mar 18 '15 at 17:49




It has the correct set of zeroes, hence the quotient is an entire analytic function
– Hagen von Eitzen
Mar 18 '15 at 17:49










2 Answers
2






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up vote
15
down vote



accepted










Using the trig identity



$$sin (2t) = 2sin (t) cos (t),$$



we have



$$prod_{n = 1}^N cos(x/2^n) = prod_{n = 1}^N frac{sin(x/2^{n-1})}{2sin(x/2^n)} = frac{sin(x)}{2^Nsin(x/2^N)} = frac{sin x}{x}cdot frac{x/2^N}{sin(x/2^N)}$$



Take the limit as $N to infty$ and use the fact $lim_{tto 0} frac{sin t}{t} = 1$ to obtain the result.






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  • I'm not the OP, so feel free to ignore this request, but I'm lost on the argument that $lim_{Ntoinfty} 2^N sin(2^{-N}x) = x$
    – jameselmore
    Mar 18 '15 at 17:59












  • @jameselmore By the Taylor expansion we have $sin(2^{-N}x)sim_{Ntoinfty}2^{-N}x$.
    – user63181
    Mar 18 '15 at 18:02










  • @jameselmore I've added more details. Since $(sin t)/t to 1$ as $t to 0$, $(sin x/2^N)/(x/2^N) to 1$ as $N to infty$. Therefore $(x/2^N)/sin(x/2^N) to 1$ as $N to infty$.
    – kobe
    Mar 18 '15 at 18:04










  • @kobe, Thanks! Makes a lot of sense
    – jameselmore
    Mar 18 '15 at 18:06


















up vote
5
down vote













Hint



$$cos(x/2^n)=frac12frac{sin(x/2^{n-1})}{sin(x/2^n)}$$
and telescope.






share|cite|improve this answer





















  • How do you incorporate the denominator $x$ into your proof?
    – DeepSea
    Mar 18 '15 at 17:51











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
15
down vote



accepted










Using the trig identity



$$sin (2t) = 2sin (t) cos (t),$$



we have



$$prod_{n = 1}^N cos(x/2^n) = prod_{n = 1}^N frac{sin(x/2^{n-1})}{2sin(x/2^n)} = frac{sin(x)}{2^Nsin(x/2^N)} = frac{sin x}{x}cdot frac{x/2^N}{sin(x/2^N)}$$



Take the limit as $N to infty$ and use the fact $lim_{tto 0} frac{sin t}{t} = 1$ to obtain the result.






share|cite|improve this answer























  • I'm not the OP, so feel free to ignore this request, but I'm lost on the argument that $lim_{Ntoinfty} 2^N sin(2^{-N}x) = x$
    – jameselmore
    Mar 18 '15 at 17:59












  • @jameselmore By the Taylor expansion we have $sin(2^{-N}x)sim_{Ntoinfty}2^{-N}x$.
    – user63181
    Mar 18 '15 at 18:02










  • @jameselmore I've added more details. Since $(sin t)/t to 1$ as $t to 0$, $(sin x/2^N)/(x/2^N) to 1$ as $N to infty$. Therefore $(x/2^N)/sin(x/2^N) to 1$ as $N to infty$.
    – kobe
    Mar 18 '15 at 18:04










  • @kobe, Thanks! Makes a lot of sense
    – jameselmore
    Mar 18 '15 at 18:06















up vote
15
down vote



accepted










Using the trig identity



$$sin (2t) = 2sin (t) cos (t),$$



we have



$$prod_{n = 1}^N cos(x/2^n) = prod_{n = 1}^N frac{sin(x/2^{n-1})}{2sin(x/2^n)} = frac{sin(x)}{2^Nsin(x/2^N)} = frac{sin x}{x}cdot frac{x/2^N}{sin(x/2^N)}$$



Take the limit as $N to infty$ and use the fact $lim_{tto 0} frac{sin t}{t} = 1$ to obtain the result.






share|cite|improve this answer























  • I'm not the OP, so feel free to ignore this request, but I'm lost on the argument that $lim_{Ntoinfty} 2^N sin(2^{-N}x) = x$
    – jameselmore
    Mar 18 '15 at 17:59












  • @jameselmore By the Taylor expansion we have $sin(2^{-N}x)sim_{Ntoinfty}2^{-N}x$.
    – user63181
    Mar 18 '15 at 18:02










  • @jameselmore I've added more details. Since $(sin t)/t to 1$ as $t to 0$, $(sin x/2^N)/(x/2^N) to 1$ as $N to infty$. Therefore $(x/2^N)/sin(x/2^N) to 1$ as $N to infty$.
    – kobe
    Mar 18 '15 at 18:04










  • @kobe, Thanks! Makes a lot of sense
    – jameselmore
    Mar 18 '15 at 18:06













up vote
15
down vote



accepted







up vote
15
down vote



accepted






Using the trig identity



$$sin (2t) = 2sin (t) cos (t),$$



we have



$$prod_{n = 1}^N cos(x/2^n) = prod_{n = 1}^N frac{sin(x/2^{n-1})}{2sin(x/2^n)} = frac{sin(x)}{2^Nsin(x/2^N)} = frac{sin x}{x}cdot frac{x/2^N}{sin(x/2^N)}$$



Take the limit as $N to infty$ and use the fact $lim_{tto 0} frac{sin t}{t} = 1$ to obtain the result.






share|cite|improve this answer














Using the trig identity



$$sin (2t) = 2sin (t) cos (t),$$



we have



$$prod_{n = 1}^N cos(x/2^n) = prod_{n = 1}^N frac{sin(x/2^{n-1})}{2sin(x/2^n)} = frac{sin(x)}{2^Nsin(x/2^N)} = frac{sin x}{x}cdot frac{x/2^N}{sin(x/2^N)}$$



Take the limit as $N to infty$ and use the fact $lim_{tto 0} frac{sin t}{t} = 1$ to obtain the result.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Mar 18 '15 at 18:02

























answered Mar 18 '15 at 17:50









kobe

34.6k22247




34.6k22247












  • I'm not the OP, so feel free to ignore this request, but I'm lost on the argument that $lim_{Ntoinfty} 2^N sin(2^{-N}x) = x$
    – jameselmore
    Mar 18 '15 at 17:59












  • @jameselmore By the Taylor expansion we have $sin(2^{-N}x)sim_{Ntoinfty}2^{-N}x$.
    – user63181
    Mar 18 '15 at 18:02










  • @jameselmore I've added more details. Since $(sin t)/t to 1$ as $t to 0$, $(sin x/2^N)/(x/2^N) to 1$ as $N to infty$. Therefore $(x/2^N)/sin(x/2^N) to 1$ as $N to infty$.
    – kobe
    Mar 18 '15 at 18:04










  • @kobe, Thanks! Makes a lot of sense
    – jameselmore
    Mar 18 '15 at 18:06


















  • I'm not the OP, so feel free to ignore this request, but I'm lost on the argument that $lim_{Ntoinfty} 2^N sin(2^{-N}x) = x$
    – jameselmore
    Mar 18 '15 at 17:59












  • @jameselmore By the Taylor expansion we have $sin(2^{-N}x)sim_{Ntoinfty}2^{-N}x$.
    – user63181
    Mar 18 '15 at 18:02










  • @jameselmore I've added more details. Since $(sin t)/t to 1$ as $t to 0$, $(sin x/2^N)/(x/2^N) to 1$ as $N to infty$. Therefore $(x/2^N)/sin(x/2^N) to 1$ as $N to infty$.
    – kobe
    Mar 18 '15 at 18:04










  • @kobe, Thanks! Makes a lot of sense
    – jameselmore
    Mar 18 '15 at 18:06
















I'm not the OP, so feel free to ignore this request, but I'm lost on the argument that $lim_{Ntoinfty} 2^N sin(2^{-N}x) = x$
– jameselmore
Mar 18 '15 at 17:59






I'm not the OP, so feel free to ignore this request, but I'm lost on the argument that $lim_{Ntoinfty} 2^N sin(2^{-N}x) = x$
– jameselmore
Mar 18 '15 at 17:59














@jameselmore By the Taylor expansion we have $sin(2^{-N}x)sim_{Ntoinfty}2^{-N}x$.
– user63181
Mar 18 '15 at 18:02




@jameselmore By the Taylor expansion we have $sin(2^{-N}x)sim_{Ntoinfty}2^{-N}x$.
– user63181
Mar 18 '15 at 18:02












@jameselmore I've added more details. Since $(sin t)/t to 1$ as $t to 0$, $(sin x/2^N)/(x/2^N) to 1$ as $N to infty$. Therefore $(x/2^N)/sin(x/2^N) to 1$ as $N to infty$.
– kobe
Mar 18 '15 at 18:04




@jameselmore I've added more details. Since $(sin t)/t to 1$ as $t to 0$, $(sin x/2^N)/(x/2^N) to 1$ as $N to infty$. Therefore $(x/2^N)/sin(x/2^N) to 1$ as $N to infty$.
– kobe
Mar 18 '15 at 18:04












@kobe, Thanks! Makes a lot of sense
– jameselmore
Mar 18 '15 at 18:06




@kobe, Thanks! Makes a lot of sense
– jameselmore
Mar 18 '15 at 18:06










up vote
5
down vote













Hint



$$cos(x/2^n)=frac12frac{sin(x/2^{n-1})}{sin(x/2^n)}$$
and telescope.






share|cite|improve this answer





















  • How do you incorporate the denominator $x$ into your proof?
    – DeepSea
    Mar 18 '15 at 17:51















up vote
5
down vote













Hint



$$cos(x/2^n)=frac12frac{sin(x/2^{n-1})}{sin(x/2^n)}$$
and telescope.






share|cite|improve this answer





















  • How do you incorporate the denominator $x$ into your proof?
    – DeepSea
    Mar 18 '15 at 17:51













up vote
5
down vote










up vote
5
down vote









Hint



$$cos(x/2^n)=frac12frac{sin(x/2^{n-1})}{sin(x/2^n)}$$
and telescope.






share|cite|improve this answer












Hint



$$cos(x/2^n)=frac12frac{sin(x/2^{n-1})}{sin(x/2^n)}$$
and telescope.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 18 '15 at 17:49







user63181



















  • How do you incorporate the denominator $x$ into your proof?
    – DeepSea
    Mar 18 '15 at 17:51


















  • How do you incorporate the denominator $x$ into your proof?
    – DeepSea
    Mar 18 '15 at 17:51
















How do you incorporate the denominator $x$ into your proof?
– DeepSea
Mar 18 '15 at 17:51




How do you incorporate the denominator $x$ into your proof?
– DeepSea
Mar 18 '15 at 17:51


















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