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

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3
votes
1answer
109 views

does invertibility of product imply invertibility of each term of product?

Suppose $\mathcal{H}$ is a Hilbert space and the product $T_1T_2 \in B(\mathcal{H})$ is invertible. Does this imply that both $T_1, T_2$ are invertible ? I am unable to prove this since, unlike the ...
-2
votes
4answers
315 views

Derivative of product notation?

Presume $f(x,y)$ is a continuous function. How would I take the derivative of $$\prod_{x=1}^N f(x,y)$$? Edit: derivative with respect to $x$, that is.
1
vote
0answers
79 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
0
votes
1answer
144 views

Countable product of finite sets with a new metric, compact?

Suppose we have a finite set $E$. Is it true that $E^n$ is compact? The metric on $E^n$ is : $$d(\omega,\omega\prime)=\begin{cases}2^{-\inf \{ n \in \mathbf N:\omega _n \ne \omega'_n\} }&{\omega ...
1
vote
1answer
136 views

Prove that if $η$ is exact, then $η∧β$ is also exact.

Prove that if $η$ is exact, then $η∧β$ is also exact. Please give a clear way to solve?
1
vote
2answers
310 views

Maximize the product of linear functions

Suppose $f(x,y) = \prod_{i=1}^n (a_ix+b_iy)$ where $n$ is a constant larger than 500, and $a_i>0$, $b_i>0$ are known coefficient. There is only one global maximum. What's the most efficient ...
19
votes
3answers
458 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 $$ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$$
2
votes
1answer
78 views

proving than an infinite product defines an entire function

Consider the infinite product $$F(z)=\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$ How can i prove that $F$ is entire? Can i write $F$ as a Weierstrass product $\prod ...
1
vote
0answers
83 views

Bounding the product of a sequence

I am trying to find an upper bound for the following sequence: $$(1-p_1)(1-(p_1+p_2))\cdots(1-(p_1+\cdots+p_n))$$ with $n$ groups to multiply. I have written it like this: $$\prod_{i=1}^n \left({1 ...
3
votes
1answer
113 views

Deducing that “the probability of the intersection is (or is not) the product of the probabilities” from knowledge about other intersections

Let $A_1, A_2, \ldots, A_n$ be a collection of events in a probability space. There are $2^n - n - 1$ subsets S of $\{1, 2, \ldots, n\}$ for which we may or may not have $P(\bigcap_{j \in S}A_j) = ...
12
votes
1answer
545 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 ...
9
votes
3answers
385 views

Prove this product

How to prove this product? $$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
1
vote
0answers
55 views

Analytic Integration of product of exponential families

I'm happy to join your community and I hope you can help me solve this seemingly straightforward dilemma I am facing. For my thesis, I am trying to solve analytically a product of two distributions ...
8
votes
2answers
241 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
0
votes
1answer
193 views

Weak direct product

I am just reading the book "Algebra" by Hungerford and on one page it says that if $G_i$ is a family of groups $\forall i\in I$ then $\prod_{i\in I}^{w}G_i\unlhd\prod_{i\in I}G_i$ where ...
14
votes
5answers
765 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.
4
votes
2answers
643 views

product of two distinct squares

Is there any shorter and efficient way to find if a number can be formed by the product of two distinct square numbers for example 36=4*9 144=16*9 help me with an algorithm or the logic
2
votes
1answer
269 views

Convergence of infinite product

This could be something which is already somewhere in the website, but I am unable to locate any. Prove $$\prod_{n=1}^{\infty} (1-z^n)$$ converges absolutely and uniformly on each compact subset of ...
2
votes
4answers
148 views

The product $\prod_{m=1}^{11} (x^m - m)$

What would be the co-efficient of $x^{60}$ in the expansion of $\space$ $\prod_{m=1}^{11} (x^m - m)$ ?
1
vote
1answer
49 views

Asymptotics of a Product of Rational Expressions

The following is taken from page 8 of Alon and Spencer's The Probabilistic Method. $$ \prod_{i = 0}^{n-1} \frac{v - 2i}{v-i} \sim e^{-n^2/2v} $$ as long as $v \gg n^{3/2}$, estimating ...
5
votes
1answer
132 views

Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$

Difficult question from some test somewhere (I forget). $$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$ $x$ is, of course, an integer.
4
votes
2answers
147 views

Can $\prod\limits_{k=0}^n \left( 2 \cosh(2^kx)-1 \right)$ be simplified?

Do you know if the product $\prod\limits_{k=0}^n \left( 2 \cosh(2^kx)-1 \right)$ can be simplified?
1
vote
3answers
188 views

Why isn't every coproduct a product (and vice-versa)?

So I know that every coproduct is not a product, so I am misunderstanding some part of the definition of (co)products. Saying that U is a coproduct (the disjoint union of X1 and X2 below) of objects ...
1
vote
1answer
140 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 ...
2
votes
1answer
121 views

Limit of an n-ary product

Since a definite integral is defined as $$\lim_{n\to\infty} \sum_{i=0}^n f(x_i^*)\,\Delta x = \int_a^b f(x)\,dx$$ and the integral is much easier to calcluate than a sum, if we change the sum to a ...
4
votes
1answer
151 views

Operators - sums, products, exponents, etc.

$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$. $(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$. Is there an operator, such that if ...
12
votes
2answers
461 views

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

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 }}} } ...
1
vote
1answer
267 views

Homeomorphism of product of topological spaces

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...
4
votes
2answers
191 views

Are Euclid numbers squarefree?

Are Euclid numbers squarefree ? An Euclid number is by definition a Primorial number + 1. See http://mathworld.wolfram.com/Primorial.html. In notation the $n$ th Euclid number is written as $E_n = ...
0
votes
2answers
235 views

the limit of infinite product $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots $

I wonder if the function $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots, 0< y<1$, converges to some well-known function. If we let $ (1+y)(1+y^2)(1+y^3)(1+y^4)\cdots = \prod_{i=1}^\infty (1+y^i) = ...
3
votes
0answers
228 views

Product of sines

I am looking to evaluate $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}$$ without using complex numbers. I can show the result if $n$ is a power of $2$, but if $n$ is anything else I reach a point where I ...
1
vote
2answers
109 views

Partial derivates of product

How to derive from this formula: $$\frac{\partial(\mathbf g.\mathbf h)}{\partial \mathbf x} = \left(\frac{\partial(\mathbf g.\mathbf h)}{\partial x_1},\frac{\partial(\mathbf g.\mathbf h)}{\partial ...
1
vote
1answer
84 views

How to calculate the weight individual fractions to equal the weighting of the product of the same fractions

What is the formula to apply a weighting to 2 fractions individually to get the same answer if you weight their product? In the example below 50% * 100% = 50%. Multiplied times 80% (weighting) the ...
2
votes
2answers
226 views

Approach to limit of infinite product

I was wondering if there is any proof that the limit of infinite product $$\lim_{n \to \infty} \prod_{i=1}^{n} x_i, \mathrm{where}$$ $$0 < x_i < 1$$ is equal to 0 and that it does not ...
0
votes
3answers
446 views

Product of sum expression

I am having a little trouble following an example I came across today which says that: $$2 \sum_{k=1}^{n} \sum_{i=0}^{k-2} 1 = 2 \sum_{k=1}^{n} (k-1) = 2 \sum_{j=0}^{n-1} j$$ I have tried fidgeting ...
0
votes
1answer
103 views

A sum that is equal to a product

Let $x=(x_1,\ldots,x_n)\in[-1,1]^n$, $k=(k_1,\ldots,k_n)$ and $\mathbb{N}^n_0=\mathbb{N}^n \cup\{0,\ldots,0\}$. Define $ x^k$ by $$x^k=\prod_{i=1}^n x_i^{k_i}$$ Show that ...
2
votes
1answer
57 views

Minimizing the product of some variables with constant summation having an additional condition

What is the minimum of $a_1\times a_2 \times \dots \times a_n$ such that $a_1+a_2+\dots+a_n=S$ and $0 < x \le a_i \le (1+\alpha)\frac{S}{n}$? My conjecture is that we need to set as many ...
1
vote
0answers
40 views

What is the meaning of a surface approximation equation?

Given a set of $n$ points $P$, a point $p_i\in{P}$, $1\leq i\leq n$ and a number $k<n$, I define the group $N_k(p_i)$ as the group containing $p_i$'s $k$ nearest neighbors. In addition, each point ...
3
votes
0answers
440 views

How to avoid numerical overflow while computing a sum of products?

Suppose we have $N$ vectors $\vec{x}_1, \vec{x}_2,\dots,\vec{x}_N$. $\vec{x}_i$ is a $M$-dimensional vector: $\vec{x}_i = \left[ x_{i1}\;\; x_{i2}\;\; \dots \;\;x_{iM}\right]^T$ with all ...
9
votes
5answers
947 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, ...
0
votes
2answers
34 views

An inequality involving a product

Let $x_1\in(0,1)$ and $a_1,\ldots,a_n\ge-1$ reals. We know that \begin{equation} \prod_{i=1}^n (1+x_1a_i) < 1 \end{equation} Does it then also hold true that \begin{equation} \prod_{i=1}^n ...
3
votes
1answer
94 views

Derivative of $f(s) = \frac{1}{\prod_{k=1}^{n}(x_k-s)}$

Denote $$ f'_{1}(s) = \bigg( \frac{1}{x_1-s} \bigg)'_{s} = \frac{1}{(x_1-s)^2}\\ f'_{2}(s) = \bigg( \frac{1}{(x_1-s)(x_2 -s)} \bigg)'_{s} = \frac{x_1 +x_2 - 2s}{((x_1-s)(x_2-s))^2} $$ and so on. ...
3
votes
2answers
88 views

Calculate $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$

Please help me solving $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
3
votes
2answers
126 views

non-homomorphic algebras

On the page http://planetmath.org/encyclopedia/DirectProduct9.html I have read, that we may embed $\boldsymbol{A}$ into $\boldsymbol{A}\times \boldsymbol{B}$ if there exists homomorphism ...
5
votes
1answer
117 views

Products in the category of sets and (left-)total relations

By a total (or left-total) relation I mean a binary relation $R \subseteq X \times Y$ where there is, for each $x \in X$, at least one $y \in Y$ with $(x,y) \in R$. Equivalently stated, I mean ...
2
votes
1answer
115 views

How do you take the product of Bernoulli distribution?

I have a prior distribution, $$p(\boldsymbol\theta|\pi)=\prod\limits_{i=1}^K p(\theta_i|\pi).$$ $\theta_i$ can equal $0$ or $1$, so I am using a Bernoulli distribtion so that ...
4
votes
2answers
630 views

How to evaluate $\lim\limits_{n\to+\infty} \prod\limits_{k=1}^n (1+k/n^2)$?

I've got a limit which puzzle me several days. The question is $$ \lim_{n\to+\infty} \prod_{k=1}^n\left(1+\frac{k}{n^2}\right).$$ Can you help me? Thank you in advance
2
votes
1answer
113 views

Is there a relationship between products and integrals? Is there a way to convert a product into an integral?

I know that the Euler-Maclaurin formula establishes a relationship between sums and integrals, but is there some sort of formula that establishes a relationship between products and integrals? I don't ...
8
votes
0answers
186 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 - ...
23
votes
0answers
770 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 ...