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

learn more… | top users | synonyms

1
vote
1answer
36 views

Closed formula for finite product series

I need to solve the recurrence: $$ \begin{align*} T(n) &= kT\left(\frac{n}{2}\right) + (k - 2)n^3 \\ &\textit{where}\; k \in \mathbb{Z}: k \geq 2 \\ &= k(kT\left(\frac{n}{...
5
votes
1answer
58 views

Convergence of $\prod (1+ta_n)$ implies convergence of $\sum a_n$ and $\sum a_n^2$

Let $a_n$ be a sequence of real numbers and assume that $\prod _n(1+ta_n)$ converges for two non-zero values of $t$, say $t_1, t_2\in \mathbb R\setminus \{0, -1/a_1, \ldots, -1/a_i, \ldots \}$. ...
0
votes
1answer
29 views

Showing multiplication inequality using induction

Use induction to show that: $$\frac{1}{2}\frac{3}{4}\dots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n+1}} $$ for $n > 1$.
1
vote
1answer
57 views

Triples of natural numbers with same sum and product

Im looking at pairs of triples of natural numbers without repititions such that the sums of the two triples are equal and the products of the two triples are equal. To be precise: Let $x<y<z$ ...
3
votes
2answers
279 views

Is knowing the Sum and Product of k different natural numbers enough to find them?

Can we uniquely identify the set of k different natural numbers (no two are the same) by knowing only their sum and product (and the value of k itself)?
0
votes
1answer
32 views

Expressing a product in terms of the sum

While solving a problem, I got to the expression $$(-a+b+c)(a-b+c)(a+b-c).$$ I would like to express it in terms of the sum $a+b+c$. Is there any possibility?
2
votes
2answers
87 views

Sum of all Products on Catalan numbers

how can I simplify this? let: $$ C_n = {{2n \choose n}\over n+1} $$ find: $$ \sum_{P_1 + P_2 + ... + P_k = r} \left(\prod_{j = 1}^k C_{P_j}\right) $$ thanks!
2
votes
1answer
34 views

How can I express such a product?

I know for example that $$\prod^{k}_{n=0} a_n = a_0 \cdot a_1 \cdot a_2 \cdot a_3 \cdots a_k$$ But what if I wanted to express $\space 3^k$ as a product? I know it sounds like a simple question, ...
3
votes
1answer
42 views

What is condition that the sum of $n$ complex numbers eaquals their product

Let $n\geq2$ and let $\{z_1,\dots,z_n\}$ be a set of complex numbers. Is there a condition on the $z_i$'s such that $$\sum_{i=1}^n z_i=\prod_{i=1}^n z_i$$ is identically true? For $n=2$ the ...
1
vote
1answer
23 views

Limit of products in $x_n = 1-An^{-\alpha}$ and their summation

Suppose that we have $A >0, \alpha >0$, and for each $n$, define $x_n = 1-An^{-\alpha}$ such that for large $n$ we have $x_n \in (0,1)$. Also, define the product sequence, $y_n = \prod_{i=0}^n ...
0
votes
0answers
28 views

Which kind of product do we have here?

The following GAP-output ...
6
votes
1answer
105 views

Simplify Product of sines

Is there a way simplify this product? $$ \sin\left({n} \frac{\pi}{2}\right) \sin\left({n} \frac{\pi}{3}\right) \sin\left({n} \frac{\pi}{4}\right) ...\sin\left({n} \frac{\pi}{n-1}\right) $$ And, is ...
0
votes
1answer
20 views

Elementary proof: division by integer makes real number smaller.

It's been so long since I took discrete math, maybe someone here can help me with what ought to be a straight forward and simple proof. Effectively I want to show this: Let a and b be positive ...
3
votes
1answer
59 views

How can I prove that every finite product can be transformed to the given form?

Suppose, the permutations $a=(123)$ , $b=(12)(34)$ , $c=(12345)$ and $d=(12)(35)$ are given. I checked with GAP that the elemts $$a^jb^kc^ld^m$$ with $0\le j\le 2$ , $0\le k\le 1$ , $0\le l\le 4$ , $...
1
vote
2answers
44 views

cup product in relative cohomology; why the subsets $A$ and $B$ of $X$ have to be open?

I have a question about cup products in relative cohomology. In lecture we defined the cup product on singular cohomology as follows: Let $R$ be a commutative ring with unit $1_R$, let $X$ be a ...
5
votes
0answers
35 views

Prove $\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kx)\,dx$

Let $N>0$ and $a_0,a_1,...,a_N$ be any positive numbers. How to prove that $$\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(...
0
votes
0answers
25 views

Approximate product by product

Let $\mathbb A _n = \{a_1, \ldots, a_n\} \subset \mathbb R_+$. For given $n, K \in \mathbb N$ can we bound from above the following: $$\left|\prod _{k=1}^K x_k - \prod _{\ell=1}^Lb_\ell \right| \leq f(...
0
votes
2answers
76 views

Pseudocode of brute-force algorithm that finds largest product of two numbers in a list [closed]

This one will require a basic knowledge of some computer science concepts. I am trying to come up with a pseudocode brute-force algorithm that finds the largest product of two numbers in a list $a_1, ...
2
votes
2answers
78 views

Linear approximation to the product: $\prod_{k=0}^r\left(1+\frac12\left(\frac{\frac12+k+1}{\frac12+k}-\frac{\frac12+k}{\frac12+k+1}\right)\right)$

I have come upon with the next expression: \begin{equation} P_r=\prod_{k=0}^r \left(1+\frac{1}{2}\left(\frac{\frac{1}{2}+k+1}{\frac{1}{2}+k} -\frac{\frac{1}{2}+k}{\frac{1}{2}+k+1}\right)\right) \end{...
0
votes
1answer
30 views

Why the usage of $H$ in the dot product $x^H y$ instead of $T$ or $'$ for transpose?

Given two vectors $v, u \in \mathbb{R}^n$ (i.e. column vectors), then the dot product of them is defined like this $$\sum_{i=1}^n v_i * u_i$$ Or usually it can be expressed in matrix-product ...
2
votes
0answers
41 views

How can I find the elements generating a group in a special way?

Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le j_i\...
1
vote
3answers
38 views

Prove that $\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - \sum_{n \in \mathbb{N}}{a_n}$

The following proof is obtained from this paper. My question is how to obtain the inequality. My guess is because of the following inequality: $$\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - \sum_{...
0
votes
0answers
11 views

To clear for variable 'a' in a sum of dependent products

I can't seem to find a way to clear this equation for variable $a$: $E[k] = \displaystyle\sum_{k=1}^nk\frac{a}{n+a-k}\displaystyle\prod_{i=0}^{k-1}1-\frac{a}{n+a-i}$ Do you think it's possible? Any ...
2
votes
0answers
45 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say $(\omega^...
0
votes
0answers
35 views

speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$

I would like to find the speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$ for any $z$. For example, if $|z| < 1$, doing taylor expansion we know that it is roughly $e^{zn}.$ But I need it ...
0
votes
3answers
47 views

universal property of product: must any map satisfying it be a morphism

I am thinking about the universal property of products: Let $X$ and $Y$ be objects of a category $D$. The product of $X$ and $Y$ is an object $X \times Y$ together with two morphisms $\pi_1 : X \...
3
votes
2answers
88 views

Product of Primes

Let $\mathbb{P}$ denote the set of prime numbers. How would one evaluate $$\prod_{p\in \mathbb{P}}\frac{p-1}{p}$$ I do not think that the fact that $$\prod_{n=2}^{\infty}\frac{n-1}{n}=\lim_{n\to\infty}...
2
votes
1answer
67 views

Evaluation of $\prod^{n}_{r=1}\sin \left(\frac{(2r-1)\pi}{2n}\right)$

Find value of $$\prod^{n}_{r=1}\sin \left(\frac{\left(2r-1\right)\pi}{2n}\right)$$ Where $n\in \mathbb{N}$ and $n>1$ $\bf{My\; Try::}$ Let $$P = \sin \left(\frac{\pi}{2n}\right)\cdot \sin \left(\...
0
votes
1answer
30 views

What is the name for this product?

I have a vectors like: $\vec{a} = [a_1, a_2] $ $\vec{b} = [b_1, b_2] $ And I need a vector of products of unique combinations like: $\vec{p} = [a_1 b_1, a_1 b_2, a_2 b_1, a_2 b_2]$ does exist a ...
1
vote
1answer
58 views

Product of two sums, one finite and one infinite

I'm working on a problem and I'm not sure how to find the product of these two sums: $\left(\sum_{k=0}^{\infty}\text{something}\right)\left(\sum_{k=n}^{n}\text{something else}\right)$ The "something"...
1
vote
1answer
86 views

What symbol is used for product topology?

Let $((X_k,\tau_k))_{k \in N}$ be topological spaces. The product topology $\tau$ on $X = \prod_{k \in N} X_k$ is the coarsest topology that makes all projections $\pi_k:X \to X_k$ continuous. Is ...
0
votes
0answers
31 views

Number of optimas of product of convex functions

I am dealing with a function, which is a product of two strongly convex functions, and trying to determine the number of its local minimum. For example, I have $$H=f(x)\cdot g(x)$$, in which both $f$ ...
3
votes
1answer
41 views

Simple formula for the $n$-ary version of $(x,y) \mapsto \frac{x+y}{1-xy}$

Let $x * y = \frac{x + y}{1 - xy}$. I want a single formula for $x_1 * x_2 * \ldots * x_n$, for all natural $n$. In order to generate plausible candidates, let's see what happens at small values of $...
0
votes
0answers
74 views

Countable product of first/second countable spaces is first/second countable.

Let $N$ be a countable indexing set, and $((X_k,\tau_k))_{k \in N}$ topological spaces. Define $X = \prod_{k \in N} X_k = X^N$ and let $\tau$ be the product topology on $X$ induced by $\tau_k$s. ...
0
votes
0answers
15 views

Solving $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $

I would like to work out the result of $\Pi^t_i 2 m_i \left(N_i!\right)^{m_i} $. Here, $t, i, N_i, m_i$ are positive integers. My effort: $$ \Pi^t_i 2 m_i \left(N_i!\right)^{m_i} \implies (2 m_1 \...
0
votes
2answers
41 views

How to find no. of digits of a large product

The question is: The product of 45,454,545,454,545 and 1,234 contains how many digits? I dont know how to solve it other than typing it in my calculator, but that method is wrong too.
0
votes
2answers
41 views

Define Derivative of Product of Polynomials

I have a a problem with defining a certain term... The derivative of a product of polynomials is the sum of derivatives of the products of the summands of the polynomials of the original product. (...
0
votes
2answers
74 views

How to multiply a vector from the left side with matrix?

I have always dealt with vector - matrix multiplication where the vector is the right multiplicand, but I am not sure how to apply the product between a matrix and a vector when the vector is the left ...
1
vote
0answers
32 views

Product of Several Functions Becomes Very Small: Scaling?

I have the following ratio: $$\frac{\sum_{i = 1}^n Y_i \prod_{p = 1}^P \lambda_p^{z_{i,p}}}{\sum_{i = 1}^n \prod_{p = 1}^P \lambda_p^{z_{i,p}}}$$ where $\lambda_p \in (0,1]$ is a parameter, and $z_{...
0
votes
0answers
41 views

Is there a product integral that preserves zeroes?

The integral essentially takes the arithmetic mean of the range of a function multiplied by the domain, adding together each possible output weighted by the amount of the domain accounted for by that ...
4
votes
1answer
199 views

Properties of Weak Convergence of Probability Measures on Product Spaces

EDIT: For the Bounty, I made a substantial edit revision concerning the structure of the question, to make it more readable (hopefully). Moreover I added a question on problem 2.7 of Billingsley’s ...
1
vote
1answer
48 views

A property of product order

Let $\mathfrak{A}$ be a poset, let $a\in\mathfrak{A}$. By definition $$\star a = \{ x\in\mathfrak{A} \mid \text{there exists non-least } y\in\mathfrak{A} \text{ such that } y\le a \text{ and } y\le x\}...
2
votes
2answers
53 views

Factorial Representation of product

So I've been trying to work out if it is possible to write: $\large \Pi_{i=1}^n (3i-1)$ as an expression involving the quotient or product of two factorials, or really any expression involving ...
1
vote
0answers
20 views

Product of a matrix and a tensor

I need to know how to compute the following product: $M(x)\frac{\partial M(x)}{\partial x}M(x)$ $\quad$ where $x \in R^{n}$. Assuming the dimensions of the matrices are compatible,how do we take ...
1
vote
2answers
53 views

Prove $\prod\limits_{k=0}^{n-1} \left(x^2-2x\cos \left(\alpha+\frac{2k\pi}{n}\right)+1\right)=x^{2n}-2x^n\cos(n\alpha)+1$

I have read in a paper that there is a formula as follows: $$\prod_{k=0}^{n-1} \left(x^2-2x\cos\left(\alpha+\frac{2k\pi}{n}\right)+1\right)=x^{2n}-2x^n\cos(n\alpha)+1.$$ In the paper they said that we ...
0
votes
1answer
30 views

Simplifying a -1 term out of a finite product

I've come up with an algorithm that relies upon the value of the following product: $$Q_{k} =\prod_{n=0}^k [f(n) - 1]$$ Where $f(n) \ge 2$ and strictly increasing integer function [see note]. ...
1
vote
1answer
39 views

An inequality involving sums and products

I am curious to know whether the following holds or not. If $n_1,n_2,n_3,m_1,m_2$ are positive integers strictly greater than 1 such that $$n_1+n_2+n_3 > m_1 +m_2$$ then $$n_1n_2n_3 \geq m_1m_2.$$ ...
1
vote
0answers
28 views

Binomial square sum and product

Given $c,n\in\Bbb N$ what is the expression for $$S(n,c)=\binom{n}c^2+\binom{n-c}c^2+\dots+\binom{x}c^2$$ and $$P(n,c)=\binom{n}c^2\cdot\binom{n-c}c^2\cdot\dots\cdot\binom{x}c^2$$ where $x-c<c\leq ...
0
votes
1answer
29 views

General Notation for a Reductive Operation, such as Sum (Σ) or Product (Π)

In functional programming, people often use operations like "fold" or "reduce", to convert from a collection to a single object using a binary operation. This is analogous to the sum and product ...
2
votes
1answer
41 views

Alternative factorization of $\prod\limits^{n}_{k=1}k!^{k+1}$

Question: How can I succinctly express (using the product and sum notations) the following expression? $$n^{(n+1)}(n-1)^{(n+1)+n}(n-2)^{(n+1)+n+(n-1)}\cdot\cdot\cdot 1^{(n+1)+n+(n-1)+\cdot\cdot\cdot+2}...