For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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15
votes
2answers
2k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
13
votes
5answers
605 views

Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$

Evaluate $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$ I can't see anything in this limit , so help me please.
11
votes
2answers
676 views

Infinite products - reference needed!

I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to ...
17
votes
4answers
2k views

Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
18
votes
3answers
446 views

Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
16
votes
3answers
822 views

A “fast” way for computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $?

Which is the fastest paper-pencil approach to compute the product $$ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $$
3
votes
2answers
558 views

Proving: $\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A … \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $

$$\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A ... \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $$ I am very much inquisitive to see how this trigonometrical identity can be ...
16
votes
0answers
421 views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty ...
6
votes
2answers
668 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
9
votes
4answers
1k views

Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$

Recently, I ran across a product that seems interesting. Does anyone know how to get to the closed form: $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$ I ...
4
votes
5answers
795 views

The limit of infinite product

Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
1
vote
2answers
85 views

Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
0
votes
2answers
50 views

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate the limit

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate: $$\lim_{n\to \infty}a^{-nk}\prod ^k_{j=1}\left(a+\frac{j}{n}\right)^n$$ Clueless on this problem. Seek your help.
21
votes
2answers
569 views

Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$

I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating ...
10
votes
1answer
247 views

How to prove that $\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}(1-\frac{x^2}{n^2})$ [duplicate]

How to prove that $$\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$$ I tried it with the Taylor series of $\sin(x)$ but I failed. Is there any help?
1
vote
1answer
101 views

Infinite Product is converges

I am adding this problem since it is interesting and valuable to be verified here: Prove that the infinite product $\prod_{k=1}^{\infty}(1+u_k)$, wherein $u_k>0$, converges if ...
0
votes
1answer
191 views

Direct products in the category Rel

Please describe direct products in the category Rel.
10
votes
3answers
264 views

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?
6
votes
3answers
206 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
6
votes
1answer
218 views

Uncountable product in the category of metric spaces.

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...
4
votes
1answer
187 views

Is there a “continuous product”?

Is there a "continuous product" which is the limit of the discrete product $\Pi$, just like the integral $\int$ is the limit of summation $\sum$. Thanks!
2
votes
2answers
132 views

Product of two complementary error functions (erfc)

I believe that (i.e., it would be convenient if, and visually appears that) the product of the two complementary error functions: ...
2
votes
1answer
79 views

Direct products in subcategories

I have a several categories some of which are subcategories of others. I want to research properties of products in these categories but don't know where to start. How direct products in a category ...
2
votes
1answer
124 views

Infinite product of recursive sequence

Let $a_{n+1}=\sqrt {(a_n+a_{n-1})/2}$ and $a_0=a_1=2$, how to prove convergence of the product $a_0 a_1 a_2 a_3...a_\infty$, and possibly find its value?
2
votes
2answers
1k views

The derivative of a product of more than two functions

I'm trying to generalize the product rule to more than the product of two functions using the fact that I can treat the product of $n$-1 functions as a single one. Here is an example of what I mean: ...
1
vote
1answer
142 views

How would one discover this finite product identity?

I recently found the following finite product identity in a table of products: \begin{align} \prod_{k=0}^{n-1}\left[\sinh^2y+\sin^2\left(x+\frac{k\pi}{n}\right)\right]=2^{1-2n}(\cosh(2ny) ...
1
vote
1answer
218 views

Gluing together mathematical structures, how?

By structure, I mean that which is defined here: http://en.wikipedia.org/wiki/Structure_%28mathematical_logic%29 What I'm looking for is a way of gluing together structures so that each structure ...
0
votes
0answers
70 views

Direct products in a partially ordered category

Consider a category, whose set of objects is a poset. Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an ...
4
votes
2answers
275 views

Closed form expression for a product.

A simple method for evaluating a product is term cancellation. For example, the product $$\begin{align*} \prod_{k=2}^{n}\left(1-\frac{1}{k}\right)&=\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)\\ ...
4
votes
1answer
490 views

Dyson series and T product

One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand. $\{H(t_i)\}$ are ...
3
votes
1answer
280 views

A product identity involving the gamma function

I have reduced this problem (thanks @Mhenni) to the following (which needs to be proved): ...
3
votes
2answers
72 views

How to define this pattern as $f(n)$

Given a binary table with n bits as follows: $$\begin{array}{cccc|l} 2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ ...
2
votes
1answer
91 views

To What Extent Does the Cartesian Product for Algebraic Structures Generalize?

I admit this question is quite general. If we have a group (or perhaps some other algebraic structure) $G$, we can define the Cartesian product $G\times G$ of $G$ with itself. And then powers of $G$ ...
2
votes
1answer
84 views

Identity involving a recursive product

Here is yet another problem related to plane partitions. Given the recursive formula $$ \begin{align*} F(0)&=1,\\ F(r)&=\prod_{i=1}^r\frac{i+2r-1}{2i+r-2}F(r-1). \end{align*} $$ How can we ...
2
votes
1answer
2k views

How to efficiently compute a*b mod N

I'm trying to solve some problems on interviewstreet. For some problems they mention As the answers can be very big, output them modulo 1000000007. How can I ...
1
vote
5answers
73 views

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$ I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$ Also a tighter upper bound is appreciated.
0
votes
1answer
22 views

Continuing direct product on a subcategory

Let $F$ is a full subcategory of a category $G$, both categories having binary direct product. Is it always true that there is such a binary direct product in $G$ that it is a continuation of a ...
0
votes
1answer
72 views

Use Proof By Induction to find the product of consecutive odd integers up to $2n-1$

I'm a bit stuck on this inductive proof. I have to find what this is equal to. Product of $1 \times 3 \times 5 \times \ldots \times (2n-1)$ Starting with $i= 1$. What would be a good starting point?