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

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3
votes
2answers
125 views

Closed form for $\prod_{1 \leq i < j \leq k} (j - i)$?

Is there a closed form for $\prod_{1 \leq i < j \leq k} (j - i)$? It looks like something like a determinant of a Vandermonde matrix, but I can't seem to get it to fit.
4
votes
2answers
339 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)\\ ...
3
votes
1answer
239 views

Is $\prod_{\mathbb{R}}\mathbb{R} = \mathbb{R}^\mathbb{R}$?

(If the title is unclear, I'm looking at infinite cartesian product of $\mathbb{R}$ indexed by $\mathbb{R}$.) I thought that I had reasoned this rather well, as follows: $\mathbb{R}^\mathbb{R} = ...
2
votes
3answers
2k views

Rules for algebraically manipulating pi-notation?

I'm a bit of a novice at maths and want to learn more about algebraically manipulating likelihoods in statistics. There are a lot of equations that involve taking the product of a set of values given ...
2
votes
1answer
145 views

Reference about product of elliptic curves

I am wondering if there is some accessible reference to learn about product of elliptic curves and their 'properties'. For dimension 1, there is plenty to find. I think the dimension 2 case is done as ...
5
votes
1answer
113 views

Modulus of infinite product of complex functions

We know that for complex (entire) functions $f,g$ we have $|f(z)g(z)|=|f(z)||g(z)|$, where $|.|$ means complex modulus. What about if we have an infinite product? Is it true that $$\bigg| ...
1
vote
1answer
231 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 ...
3
votes
1answer
128 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?
4
votes
5answers
839 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?
3
votes
2answers
146 views

Simple properties of a direct product

I am working on some homework for modern algebra class. The problem I just finished seems relatively easy, but I have learned to be wary of that feeling when it comes to this material. Below are the ...
16
votes
3answers
859 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 $$ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $$
1
vote
1answer
131 views

Kronecker Product

Is this right $$\mathbf{A}\left(\mathbf{B}\otimes\mathbf{C}\right)\mathbf{D}=\left(\mathbf{A}\mathbf{B}\mathbf{D}\otimes\mathbf{C}\right)$$ Thanks in advance for your help.
1
vote
2answers
172 views

Product and Square Root Proof

Let $a_1$ and $a_2$ be positive integers and let $m = a_1 a_2$. Prove that at least one of $a_1$ or $a_2$ is at least $\sqrt m$. Disclosure: This is for a homework question, though the question is ...
7
votes
2answers
421 views

Proving an infinite product formula

I have found this formula and I am trying to prove it , but I have not any idea how to deal with it: $$e^{ax}-e^{bx} = ...
4
votes
1answer
211 views

Dyson series and T product (II)

After reading the previous posts related to the Dyson series, I have decided to open a new thread because there is something that I am still not understanding. It concerns the expression: $$ ...
5
votes
1answer
280 views

Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?

If $a_1\ge a_2 \ge a_3 \ldots $ and if $b_1,b_2,b_3\ldots$ is any rearrangement of the sequence $a_1,a_2,a_3\ldots$ then for each $N=1,2,3\ldots$ one has $$\sum^N_{n=1}\left(\prod_{i=1}^n b_i ...
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: ...
5
votes
1answer
306 views

When is $\displaystyle \prod_i \prod_j a_{i} a_{j} = \Bigl(\prod_i a_i\Bigr)^2$

In statistical mechanics, I used to use the procedure that if $a_{ij}=a_i a_j$ $$\prod_i\; \prod_j a_{i}a_{j} = \biggl(\prod_i a_i\biggr)\vphantom{\Bigr)}^2$$ However, today I noticed, $$\prod_i\; ...
13
votes
1answer
304 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 ...
2
votes
3answers
1k views

About the factors of the product of prime numbers

If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using ...
4
votes
1answer
564 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 ...
7
votes
1answer
204 views

Which is the Abel's theorem invoked in the context of convergence of this infinite product?

Motivation: As I wrote in this answer the following product is evaluated in Proposition 5 of Jean-Paul Allouche and Jeffrey Shallit's paper The ubiquitous Prouhet-Thue-Morse sequence ...
6
votes
1answer
279 views

Generalization of the series for $\frac{\pi^2}{6}$? Is there a more elementary proof?

In the same vein as: $ \frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots $ Starting with: $ \displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = ...
4
votes
1answer
265 views

What is $\prod_{k=1}^n (1-x^k)$?

I'd like to know what $$\prod_{k=1}^n (1-x^k)$$ evaluates to (assuming there is a simple closed form) and what it "is" in the context of commutative algebra (of which I knew little and recall ...
2
votes
2answers
653 views

Formula for Geometric Progression

Can someone help me understand the idea behind constructing a formula for the following: For $n\in\mathbb{N}$, $n\geq 2$, find and prove a formula for: $$\prod_{i=2}^n \left(1 - ...
1
vote
1answer
219 views

Efficient calculation of polynomial product

I have 2 polynomials $p_1(x_1,\ldots,x_n)$ and $p_2(x_1,\ldots,x_n)$, of which I have to compute the product, with a special property: The exponent of each variable is always either $0$ or $1$, where ...
3
votes
2answers
62 views

interval for a product to infinity

I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, ...
23
votes
4answers
1k views

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

The arithmetic mean of $y_i ... y_n$ 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 using ...
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 $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
3
votes
2answers
569 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 ...
18
votes
2answers
3k 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 $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
3
votes
4answers
3k 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 to ...