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

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4
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0answers
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Product-Decomposition of distributive lattices

Every nontrivial (bounded) distributive lattice arises as a direct power of a certain number of nontrivial product-irreducible (bounded) distributive lattices. My question is how this number can be ...
2
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1answer
80 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 ...
5
votes
1answer
179 views

Continuous maps from products of topological spaces

Let $X,Y,Z$ be topological spaces. It is well-known that if $F:X\times Y\to Z$ is a continuous map, we can define a map $$\overline{F}:X\to C(Y,Z) \\\overline{F}(x)(y)=F(x,y)$$ where $C(Y,Z)$ is the ...
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vote
3answers
393 views

set theoretic function, products of sets (product versus Cartesian product)

Regarding the products of functions in axiomatic set theory, two textbooks which I am reading (Halmos; Hrbacek/Jech) have said the following: "There is a natural one-to-one correspondence between ...
2
votes
2answers
186 views

Value of double product

What is $$ \prod_{i=1}^n\prod_{j=1}^{n-i}i^2+j^2 $$ ? It feels like there should be some way to simplify this or calculate it more efficiently than iterating over each of the $\sim n^2/2$ points. ...
6
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1answer
223 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 ...
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votes
1answer
313 views

How to expand this summation/product $\sum\limits_{n=0}^\infty \prod\limits_{j=0}^{n-1}\frac{\lambda_j}{\mu_{j+1}}$

Let $\mu_n = n\mu$ for $1 \le n \le 3$ and $\mu_n = 3\mu$ for $n \ge 4$. Let $\lambda_n = \lambda$ for all $n \in \mathbb{N}_0$. Define $\rho := \frac{\lambda}{\mu}$. How would I expand ...
4
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1answer
202 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!
6
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2answers
811 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 ...
1
vote
1answer
131 views

Problems with infinite $\Omega$, when trying to define product spaces of discrete probability spaces

Definitions In our course we defined a discrete probability space as a tuple $\left(\Omega,P\right)$, where $P:\mathcal{P}(\Omega)\rightarrow\left[0,1\right]$ and $\Omega$ is at most countable, such ...
3
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2answers
305 views

Given $\sum |a_n|^2$ converges and $a_n \neq -1$, show that $\prod (1+a_n)$ converges to a non-zero limit implies $\sum a_n$ converges.

I have been working on this problem for a while and cannot seem to make any progress without coming up with something wrong or hitting a dead end. Here is what I have so far: $ \prod (1+a_n) < ...
2
votes
1answer
221 views

Show that $\prod_{i=1}^n a_i- \prod_{j=1}^n b_i =$ $\sum_{t=1}^{n-1}(\prod_{i\leq t-1}a_i)(\prod_{j\geq t+1} b_j)(a_t-b_t)$

Pardon the cryptic notation and possibly trivial question. I believe the following holds. Define $$X_t=(\prod_{i\leq t-1}a_i)(\prod_{j\geq t+1} b_j)(a_t-b_t).$$ Show that ...
3
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0answers
169 views

Infinity Product Equality.

Let $\{I_n\}_{n\in\mathbb{N}}$ be a sequence of intervals in the form $$ I_n = \Big [ \frac{q_n}{b_n}, \frac{q_n + 1}{b_n} \Big),$$ where $q_{n}$ is some integer, for all $n\in\mathbb{N}$. Define ...
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
350 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
245 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
114 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
233 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?
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5answers
851 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?
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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
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3answers
861 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
134 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.
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2answers
174 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
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2answers
426 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
213 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 ...
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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
308 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\; ...
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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
585 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
207 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
662 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
222 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, ...
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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 ...
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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
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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
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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 ...