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

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8
votes
0answers
182 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 - ...
19
votes
0answers
619 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 ...
0
votes
1answer
219 views

Direct products in the category Rel

Please describe direct products in the category Rel.
5
votes
2answers
219 views

Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$

$f(x)=\sum\limits_{k = 2 }^ \infty e^{-kx} \ln(k) $ $\int\limits_0^{\infty}\int\limits_x^{\infty}\, f(\gamma)\, d\gamma dx=\sum\limits_{k = 2 }^ \infty \frac{1}{k^2} \ln(k) $ ...
6
votes
3answers
133 views

closed-form expressions for product of 3n+k where k = 1 or 2

There are some easy products that can be written in closed form in terms of factorials: $ 2 \times 4 \times 6 \times ... 2n = n! \times 2^n$ $ 1 \times 3 \times 5 \times ... (2n-1) = {{(2n)!} ...
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 ...
1
vote
2answers
82 views

Special dot-product

I have been wondering if the following dot product definition for the $n$-coordinate vectors $a$ and $b$ has a name: $$<a\backslash b> = \sum_{i=1}^{n} a_i*b_{n-i+1},$$ rather than the classical ...
3
votes
1answer
345 views

Prove that the dihedral group $D_4$ can not be written as a direct product of two groups

I like to know why the dihedral group $D_4$ can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, ...
10
votes
7answers
582 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.
21
votes
2answers
603 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 ...
-6
votes
1answer
208 views

Order of products and order of multipliers

I asked this question (and have received an answer) at MathOverflow. Now a little more difficult question: Let $f$ and $g$ are binary relations (on some set $\mho$). The function $f\times^{C} g$ is ...
4
votes
2answers
179 views

When equal products imply equal factors?

Under which additional conditions $a\times b = c\times d \Rightarrow a=c\wedge b=d$ (where $\times$ is a categorical product)? For example, in the case of Cartesian product, for this is enough when ...
7
votes
1answer
515 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 ...
12
votes
2answers
799 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 ...
7
votes
1answer
136 views

Zeros in the complex plane and convergence

I'm doing some number theory which requires some work in $\mathbb{C}$, but unfortunately my complex analysis is a little rusty. A text I am reading states the following: ...and given that ...
2
votes
1answer
3k 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
1answer
179 views

simplifying a product formula (similar to Euler's sine product)

Can anyone help me out trying to simplify the left hand side of the below equation to obtain the right hand side? $$ \displaystyle\prod_{\substack{n=-\infty \\n\neq ...
1
vote
3answers
562 views

Cross product and dot product

What's the easiest way to understand and prove that $A \cdot B \times C = C \cdot A \times B $ ?
0
votes
1answer
189 views

Definition of product of uniform spaces

In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous. But Springer's encyclopedia ...
4
votes
0answers
95 views

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
votes
1answer
81 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
185 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 ...
1
vote
3answers
397 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
208 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
votes
1answer
224 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 ...
0
votes
1answer
329 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
votes
1answer
205 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
votes
2answers
836 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
133 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
votes
2answers
307 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
224 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
votes
0answers
171 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
382 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
253 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} = ...
3
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
147 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
115 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
237 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
133 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
875 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
873 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
135 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
3answers
185 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
429 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
221 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
282 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
310 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\; ...