Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This question already has an answer here:

Find the limit $$\lim_{n \to \infty}\left[\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right)\right]$$

I take log and get $$\lim_{n \to \infty}\sum_{k=2}^{n} \log\left(1-\frac{1}{k^2}\right)$$

share|cite|improve this question

marked as duplicate by Normal Human, Marconius, Cameron Williams, BLAZE, Kevin Dong Nov 19 at 4:39

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

See also the near-duplicate: – Martin Feb 16 '13 at 5:53

3 Answers 3

up vote 25 down vote accepted

The identity $k^2-1=(k+1)(k-1)$ shows that $$ \prod_{k=2}^n\left(1-\frac{1}{k^2}\right)=\prod_{k=2}^n\frac{k^2-1}{k^2}=\prod_{k=2}^n\frac{k-1}k\cdot\prod_{k=2}^n\frac{k+1}k=\frac1n\cdot\frac{n+1}2, $$ and the value of limit should follow.

share|cite|improve this answer
We can call it a "telescoping product". – GEdgar Jan 25 '13 at 18:45
@GEdgar Absolutely. One probably applies telescopic more often to sums but this fits perfectly products as well. – Did Jan 25 '13 at 20:07

The $r$th term is $\frac{r^2-1}{r^2}=\frac{(r-1)}r\frac{(r+1)}r$

So, the product of $n$ terms is $$\frac{3.1}{2^2}\frac{4.2}{3^2}\frac{5.3}{4^2}\cdots \frac{(n-1)(n-3)}{(n-2)^2}\frac{(n-2)n}{(n-1)^2}\frac{(n-1)(n+1)}{n^2}$$ $$=\frac12\frac32\frac23\frac43\cdots\frac{n-2}{n-1}\frac n{n-1}\frac{n-1}n\frac{n+1}n=\frac12\frac{(n+1)}n$$

as the 1st half of any term is cancelled by the last half of the previous term except for the 1st & the last term.

share|cite|improve this answer

Your way works nice if you employ the Euler's infinite product for the sine function. Then

$$\lim_{n \to \infty}\sum_{k=2}^{n} \ln\left(1-\frac{1}{k^2}\right)=\lim_{x\to\pi}\ln\left(\frac{\pi^2\sin x}{x(\pi^2-x^2)}\right)=\lim_{y\to0}\ln\left(\frac{\pi^2\sin y}{y(2\pi-y)(\pi-y)}\right)=\ln(1/2)$$ Thus, your product is $1/2$.

share|cite|improve this answer
To Chris,s sister would you like to explain me Infinite product of sine function and how can i write $\displaystyle \lim_{n \to \infty}\sum_{k=2}^{n} \log\left(1-\frac{1}{k^2}\right)=\lim_{x\to\pi}\ln\left(\frac{\pi^2\sin x}{x(\pi^2-x^2)}\right)$ Thanks – juantheron Jan 25 '13 at 18:58
@juantheron: I simply used Euler's infinite product for the sine function, took log of both sides and then rearranged things to get the expression you see after the first equal sign. Then it remained to take the limit to $\pi$ to get our limit. For more details on infinite product of sine function, see here: – OFFSHARING Jan 25 '13 at 19:38
Thanks Chris's sister – juantheron Jan 25 '13 at 21:04
@juantheron: welcome! – OFFSHARING Jan 25 '13 at 21:06

Not the answer you're looking for? Browse other questions tagged or ask your own question.