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

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77
votes
10answers
8k views

Is $0! = 1$ because there is only one way to do nothing?

The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or ...
56
votes
3answers
4k views

Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots ...
55
votes
0answers
2k 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 $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty ...
32
votes
4answers
1k views

Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
28
votes
4answers
2k views

What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i \ldots y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and ...
27
votes
3answers
6k views

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

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
25
votes
11answers
2k views

Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition? For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b \text{-...
25
votes
4answers
4k 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 $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
23
votes
7answers
787 views

Product of cosines: $ \prod_{r=1}^{7} \cos \left(\frac{r\pi}{15}\right) $

Evaluate $$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\...
23
votes
4answers
830 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: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ $$\sum_{j=2}^\...
23
votes
1answer
653 views

Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$

If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then $$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$ This identity follows from a probability calculation ...
21
votes
4answers
2k views

How to prove those “curious identities”?

How to prove $$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$ and $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$
21
votes
2answers
699 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 ...
20
votes
3answers
576 views

Finding $ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$

I would appreciate if somebody could help me with the following problem. How can we find the product $$ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$$
20
votes
5answers
10k views

Why is there no explicit formula for the factorial?

I am somewhat new to summations and products, but I know that the sum of the first positive n integers is given by: $$\sum_{k=1}^n k = \frac{n(n+1)}{2} = \frac{n^2+n}{2}$$ However, I know that no ...
17
votes
3answers
995 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 $$ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $$
17
votes
3answers
472 views

$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$

How can I find the following product using elementary trigonometry? $$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$ I have tried using a substitution, but nothing ...
17
votes
5answers
1k views

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

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.
16
votes
2answers
536 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$ [duplicate]

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
16
votes
2answers
324 views

How to compute the following integral in $n$ variables?

How can the following integral be calculated: $$ I_n=\int_0^1\int_0^1\cdots\int_0^1\frac{\prod_{k=1}^{n}\left(\frac{1-x_k}{1+x_k}\right)}{1-\prod_{k=1}^{n}x_k}dx_1\cdots dx_{n-1}dx_n $$ There should ...
15
votes
2answers
1k 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 ...
15
votes
1answer
560 views

Prove that the product of some numbers between perfect squares is $2k^2$

Here's a question I've recently come up with: Prove that for every natural $x$, we can find arbitrary number of integers in the interval $[x^2,(x+1)^2]$ so that their product is in the form of $...
14
votes
4answers
2k 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 ...
14
votes
1answer
362 views

A question about $\prod_{x\in \mathbb{R}^{*}}{x}$

When I was an elementary school student, I encountered some puzzles like "What's product of numbers of hair of each people in the world?", which answer was zero, because there's people with no hair(...
12
votes
1answer
483 views

New Year Combinatorics 2015

In the spirit of the festive period and in appreciation of the encouraging response to my X'mas Combinatorics problem posted recently, here's one for the New Year! Express the following as a ...
12
votes
1answer
584 views

How does one calculate the product of $\tan 1^{\circ} … \tan 45^{\circ}?$

I have seen a question asked on yahoo asking to find the value of $\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \dots \cdot \tan 45^{\circ}$ (in degrees) I have seen various results concerning ...
11
votes
2answers
273 views

Is $ \prod\limits_{k=0}^\infty \left(1 + \frac{1}{k!}\right) = \mathrm e^2 $?

I was playing around and I came up with this product, which I believe to be equal to $\mathrm e^2$. $$ \prod_{k=0}^\infty \left(1 + \frac{1}{k!}\right) \stackrel{?}{=} \mathrm e^2 $$ After ...
11
votes
2answers
294 views

Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

The current issue (vol. 120, no. 6) of the American Mathematical Monthly has a proof by probabilistic means that $$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k} $$ for ...
11
votes
3answers
317 views

What is the probability that the product of $20$ random numbers between $1$ and $2$ is greater than $10000$?

Twenty random real numbers $a_1,a_2,\dots,a_{20}$ are chosen such that $1\le a_i \le 2$. What is the probability that their product is greater than $10000$? (By random, I mean each real number in the ...
11
votes
1answer
747 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?
10
votes
7answers
603 views

Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?

I am trying to construct a sequence $\{x_{n}\} \in (0,1)$ such that such that the product of all its terms is $\frac{1}{2}$. Please can I have any clue to solve my problem? Thanks.
10
votes
3answers
437 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?
10
votes
4answers
7k views

Product of two cyclic groups is cyclic iff their orders are co-prime

Say you have two groups $G = \langle g \rangle$ with order $n$ and $H = \langle h \rangle$ with order $m$. Then the product $G \times H$ is a cyclic group if and only if $\gcd(n,m)=1$. I can't seem ...
9
votes
3answers
402 views

Prove this product

How to prove this product? $$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
9
votes
5answers
2k views

Definition of the Infinite Cartesian Product

(1) If $X$ and $Y$ are two sets, we define the Cartesian product $X \times Y$ as the set of ordered pairs $(x,y)$, such that $x \in X$ and $y \in Y$. (2) On the other hand [Folland, Real Analysis, ...
9
votes
2answers
521 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm $$S=...
9
votes
1answer
219 views

How many numbers $ N \le 10^{10}$ are the product of $3$ distinct primes?

How many numbers $ N \le10^{10}$ are the product of $3$ distinct primes? I can realistically calculate any $\pi(n), n < 10^{15} $ but I don't think it's possible to list all primes $>10^8$ in ...
9
votes
2answers
266 views

How to find finite trigonometric products

I wonder how to prove ? $$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$ give me a tip
9
votes
1answer
82 views

Prove that a given expression is always an integer [duplicate]

Given integers $x_1, x_2, \dotsc, x_n$, prove that the expression $$ \prod \limits_{1\leq i<j\leq n}\frac{x_i - x_j}{i-j} $$ is always an integer. I think induction should work, but I ...
9
votes
0answers
373 views

Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
8
votes
1answer
1k views

Evaluation of a product of sines [duplicate]

Possible Duplicate: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I am looking for a closed form for this product of sines: \begin{equation} \sin \left(\frac{\pi}{n}\...
8
votes
9answers
262 views

Why is empty product defined to be $1$? [duplicate]

For example $\prod_{2 \le j < 1} 2^j= 1.$ How does that happen?
8
votes
2answers
249 views

Showing an indentity with a cyclic sum

Let $n\geqslant2$, and $k\in \mathbb{N}$ Let $z_1,z_2,..,z_n$ be distinct complex numbers Prove that $$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j \ne ...
8
votes
1answer
355 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!
8
votes
1answer
3k views

Product of sets and supremum

Let $A$ and $B$ be nonempty sets of positive real numbers that are bounded above. Also let $AB = \{ab: a \in A, b \in B \}$. Prove that $AB$ is bounded above and $\sup(AB) = (\sup A) (\sup B)$. So $\...
8
votes
1answer
309 views

Why is this sum equal to $0$?

While solving a differential equation problem involving power series, I stumbled upon a sum (below) that seemed to be always equal to $0$, for any positive integer $s$. $$ \sum_{k=0}^s \left( \frac{ \...
8
votes
2answers
460 views

Olympiad problem: Erdos-Selfridge

The following problem is a special case of Erdos-Selfridge theorem: http://projecteuclid.org/euclid.ijm/1256050816 Problem: Prove that for any positive integer $n$, the product $(n+1)(n+2)...(n+10)$ ...
8
votes
1answer
310 views

Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose m}\left(p-...
8
votes
0answers
206 views

How to find the the product $\left(1 - \frac{1}{a}\right)\left(1-\frac{1}{a^{2}}\right)\left(1-\frac{1}{a^{3}}\right)\ldots$ [duplicate]

Possible Duplicate: Result of the product $0.9 \times 0.99 \times 0.999 \times \dots$ How to find the product $$\left(1 - \frac{1}{a}\right)\left(1-\frac{1}{a^{2}}\right)\left(1-\frac{1}{a^{3}}...
7
votes
4answers
144 views

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to? As far as I can tell, this has no closed-form solution (not saying much, I don't know much math), but a friend of mine swears he saw a ...