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

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5
votes
1answer
75 views

Being $N$ a constant $>0$, show $\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this paper might be ...
5
votes
2answers
88 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
2
votes
0answers
93 views

Is $\frac{1}{e^\gamma\log x} \prod\limits_{p < x,p\,\text{prime}} \frac{p}{p-1}<1+ \prod\limits_{p<x,p\,\text{prime}}\frac{1}{p^{n+1}-1}?$

Let $n$ be an initially arbitrarily large variable, but always decreasing (and more specifically non-increasing) to exactly $1$ when $p$ is the largest prime in the product. Then, denoting with ...
0
votes
2answers
80 views

What's the name of this strange inequality?

There is an inequality: $$\sqrt[n]{\prod_{i = 1}^{n}{(a_i+b_i)}} \geq \sqrt[n]{\prod_{i = 1}^{n}{a_i}} + \sqrt[n]{\prod_{i = 1}^{n}{b_i}}$$ which I even don't know its name. I'd like to have an ask ...
3
votes
0answers
35 views

How I can calculate this product

How I can calculate this product: $$s=∏_{j=1}^{p-3}\sinh^2[2^{j-1}\cosh^{-1}( 2)]$$ for a natural number $p>3$.
0
votes
2answers
27 views

$S$ nonempty finite subset of group $G$ for which $SS=S$. $S$ is subgroup.

Let $S\subset G$, $S$ finite and nonempty, $G$ group. Suppose additionally that $$SS=\{s_1 s_2: s_1\in S, s_2 \in S\}=S.$$ How can I prove that $S$ is a subgroup of $G$? Does this hold for $S$ ...
1
vote
1answer
46 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
6
votes
9answers
74 views

Why is empty product defined to be $1$? [duplicate]

For example $\prod_{2 \le j < 1} 2^j= 1.$ How does that happen?
2
votes
0answers
58 views

Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true. (in other word for every question ...
2
votes
2answers
89 views

Idea for primality testing based on a trigonometric product

This is an idea that I had about 3 months ago. I tried some college professors, they didn't care. I tried to solve, but with no luck. I ask for help to find the closed form of the following product ...
5
votes
2answers
101 views

Characterize the type of sequence that satisfies $\prod (1-a_i) \leq c$

Consider a product $\prod_{i=1}^{n} (1-a_i)$ where $n\leq \infty$ and $a_i\in [0,1)$ for all $i$. I'm hoping to see if there exist conditions on the sequence $\{a_i\}$ so that $$\prod_{i=1}^{n} ...
0
votes
1answer
28 views

Recurrence relation for the coefficients of the polynomial $p_n(x) = \prod_{i=0}^{n-1}(x-i)$

Let's consider the polynomials $$ p_n(x) = \prod_{i=0}^{n-1}(x-i)=\sum_{i=1}^{n} a_{n,i}x^i$$. for all $n \in \mathbb{N}$. If $n=1$, then $p_1(x) = x$ and $a_{1,1} = 1$. Since I know that: ...
0
votes
1answer
23 views

Theorem? For any sets A, B, C, and D, if A x B is a subset of C x D then A is a subset of C and B is a subset D.

  Is the following proof correct? If so, what proof strategies does it use? If not, can it be fixed? Is the theorem correct?   Proof. Suppose A x B is a subset of C x D. Let a be an arbitrary element ...
1
vote
1answer
55 views

The limit of products of the form $(n^3-1)/(n^3+1)$

Calculate $$\lim_{n \to \infty} \frac{2^3-1}{2^3+1}\times \frac{3^3-1}{3^3+1}\times \cdots \times\frac{n^3-1}{n^3+1}$$ No idea how to even start.
9
votes
1answer
133 views

How to compute the following integral in $n$ variables?

How can the following integral be calculated: $$ I_n=\int_0^1\int_0^1\cdots\int_0^1\frac{\prod_{k=1}^{n}\left(\frac{1-x_k}{1+x_k}\right)}{1-\prod_{k=1}^{n}x_k}dx_1\cdots dx_{n-1}dx_n $$ There should ...
3
votes
1answer
34 views

How to calculate the product of a set

How can you calculate the product of a set $A$, denoted by $\Pi A$ and defined by $\forall z \in \Pi A(z \subseteq \bigcup A \wedge \forall y \in A (\exists x (z \cap y = \lbrace x \rbrace))) $ ...
2
votes
1answer
42 views

Product of ergodic transformations

I'm asked to give an example, that the product of two ergodic systems is not ergodic in general. I know that for $X_1=X_2=(S^1,B,m,R_a)$ (the irrational rotation on the unit circle with Lebesgue ...
1
vote
2answers
26 views

relationship between multiplication and correlation

is there a deep interpretation of multiplication as correlation? is this in some sense the most fundamental way to "combine" objects (eg numbers) into products? my reasons for asking are that the ...
1
vote
0answers
50 views

Simplify the product of two sums

How can I simplify the following product of two sums: $$ \biggl(\, \sum ^{n}_{k=0}a_{k}\biggr) \biggl(\, \sum ^{n}_{k=0}\dfrac {1}{a_{k}}\biggr) $$
3
votes
1answer
16 views

Product rule for Hessian matrix

Let $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$. Is there a general formula for the Hessian matrix of their product? That is, what is $H(f(x) g(x))$, where $H(f(x)) = ...
0
votes
1answer
34 views

Using induction to prove that $ \prod_{i=1}^{n} (1+a_{i}) \geq 1 + \sum_{i=1}^{n}a_{i} $ [closed]

I started a course in my university and I am having trouble with answering this question: Prove using Mathematical induction, for every real, non-negative 'n' number $$(a_{i}\geq 0)$$ the ...
0
votes
0answers
21 views

Product of dot products of two vectors

I have a product of innerproduct/dot product of two vectors. $ \langle u_i,v_j \rangle\cdot\langle x_i,y_j\rangle$. Is there any transformation/decomposition such that I can combine $u_i$ with $x_i$ ...
-1
votes
3answers
28 views

Product of inner products

Is product of innerproduct again a inner product of two vectors? For example - Is $ (< u,v >)(< x,y >) = < m,n > $? And if yes is m and n unique and how do we calculate those?
3
votes
1answer
78 views

Resemblance between product and homotopy

The notion of product $X\times X$ for an object $X$ of a category $C$ resembles the notion of homotopy between two continuous functions. Indeed the relevant diagrams look the same: ...
0
votes
1answer
57 views

Definition of a coproduct and its universal property - connection?

I have a problem connecting the definition of a coproduct with its often mentionend universal property. Let's start with the definition (just for two objects): Let $A_1$ and $A_2$ be objects of a ...
1
vote
0answers
38 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
0
votes
0answers
64 views

Help in writing a nasty expression in nice closed form

This question is abouting re-writing a product in nice closed form. I have the following $$f(v_1) = \left(\sum_{i=1}^K \pi \lambda_i \delta_1 v_1^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i ...
1
vote
3answers
25 views

summation and product of sin and cos

I wonder how to find summation for $\displaystyle \sum_{k=0}^{n-1}(\cos{\frac{2\pi k}{n}+i \sin\frac{2\pi k}{n}})$ and the same for product $\displaystyle \prod_{k=0}^{n-1}(cos{\frac{2\pi k}{n}+i ...
1
vote
1answer
78 views

Product of roots of unity

Does somebody have a nice proof of the following? $$\prod_{m=1}^{n-1} \frac{e^{2\pi i k m/n} - 1}{e^{2 \pi i m / n} - 1} = \begin{cases} 1 & \text{ if $\gcd(k, n) = 1$} \\ 0 & ...
1
vote
0answers
25 views

Product notation $\prod$ when product does not commute [duplicate]

This is kind of a dubious question, but is the product notation $\prod$ often used in noncommutative rings? For example, if $M_i$ are matrices, I guess the common definition of $\prod$ is $$\prod_i ...
0
votes
0answers
33 views

Minimizing sum of weighted product

Consider a total of $d$ items, $\{I_1,I_2, \cdots,I_d\}$, each having a weight $w_i$ (a positive integer), and a total of $m$ bins, $\{B_1,B_2,⋯,B_m\}$. We would like to distribute the items into the ...
0
votes
0answers
25 views

writing sum as a product and vice versa.

$\Pi = k$ from k = 1 to n Can you write this in form of sigma? So that you can evaluate it as a sum? Also, are there any shorthand formula to evaluate a product like there are for summations? ...
1
vote
2answers
31 views

Proofs of identity for product of binomial coefficients

While verifying my answer to another question, I came across a problem of binomial coefficients: Does $\hspace{.2cm}\displaystyle \prod_{k=1}^{n-1}\binom{n-1}{k}=\prod_{k=1}^{n-1}k^{2k-n}$ for all ...
4
votes
2answers
71 views

Show that $k^a=\sum_{m=1}^b\left ( c_m^a\prod_{n\neq m} \frac{k-c_n}{c_m-c_n} \right ).$

I used the following result in another post without providing proof (because I couldn't prove it): $$k^a=\sum_{m=1}^b\left ( c_m^a\prod_{n\neq m} \frac{k-c_n}{c_m-c_n} \right ),$$ where $a$ and $b$ ...
0
votes
1answer
43 views

What does $ \prod_{i = 2}^{ n-1} \frac{1}{i}$ converge to?

What does $ \prod_{i = 2}^{ n-1} \frac{1}{i}$ converge to? It boils down to $\frac{1}{2} * \frac{1}{3} * \frac{1}{4} * ... * \frac{1}{n-1}$ But is there a direct formula that gives me the same ...
8
votes
1answer
192 views

Olympiad problem: Erdos-Selfridge

The following problem is a special case of Erdos-Selfridge theorem: http://projecteuclid.org/euclid.ijm/1256050816 Problem: Prove that for any positive integer $n$, the product $(n+1)(n+2)...(n+10)$ ...
0
votes
2answers
63 views

Minimizing sum of products

Consider a total of $d$ items, $\{I_1, I_2, \cdots, I_d \}$, each having a weight $w_i$, and a total of $m$ bins, $\{B_1, B_2, \cdots, B_m\}$. We would like to distribute the items into the bins such ...
3
votes
1answer
72 views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)…$

Find a closed form for the value of the infinite product $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$ I know it converges. At first I thought it was the Euler–Mascheroni constant, but it's only accurate to about ...
1
vote
1answer
45 views

Infinite Products — Tangent function?

I've been looking around and I see no formulas given in any of the sources I've been able to find for the infinite product representing $\tan\left(x\right)$. Is it simply the ratio of the infinite ...
2
votes
2answers
470 views

Sum of real numbers that multiply to 1

I've seen a question in my math book with this explanation above it: "If the product of n positive real numbers is 1, then the sum of these numbers must be more than n". I was wondering if this is ...
1
vote
1answer
27 views

Maximum product for multisets with same sum

Given a positive number N, among all multisets (containing only positive numbers) with sum N, is there a reliable method for determining the set with the maximum product? For example, for N = 5, the ...
1
vote
1answer
21 views

Product rule trig

This was given as a solution to a question and I've tried working it out but can never get the same answer. Here $x=rcosϕ$ and $y=rsinϕ$ It's mostly the first 2 lines I don't understand. Wouldn't ...
0
votes
0answers
41 views

A product of logarithmic integrals $\displaystyle\prod_{u=1}^m\dfrac{\text{Li}_{2u+1} ​(k)}{\text{Li}_{2u}(k)}$

Let $m,k\in\mathbb{N}^*,\displaystyle P_k=\prod_{u=1}^m\dfrac{\text{Li}_{2u+1}‌​(k)}{\text{Li}_{2u}(k)}$ Is there a way to simplify this product ? What is its behavior for $k\rightarrow\infty$, for ...
3
votes
2answers
82 views

Prove that $\prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1$

Prove that $$ \prod_{k=0}^{n-1}(z-\mathrm{e}^{2k\pi i/n})=z^n-1. $$ In my some problem I have used $$ \prod_{k=0}^{7}(z-\mathrm{e}^{2k\pi i/8})=z^8-1. $$ I have verified this. So I think in general ...
2
votes
3answers
38 views

How to show that these products are equal?

I need your help. I'm trying to show that these products are equal: $$\prod_{k=1}^{n}(4k-2)=\prod_{k=1}^{n}(n+k)$$ Thank you in advance ! PS: I need two different ways to solve the problem...
0
votes
1answer
35 views

Product Integral

What is the product integral of $(1+x)^{-(\theta+1)/\theta}$, if we consider that the product integral is from x=0 to x=n? It's easy to solve 1/theta, however, the second part is a little more ...
0
votes
1answer
32 views

In how many ways can you paint 90 distinct buckets?

In how many ways can you paint 90 distinct buckets, if 25 of them must be painted red, 40 of them must be painted blue, and 25 of them must be painted green? I am right to assume that these object ...
1
vote
2answers
42 views

How many ways can the school choose a President Vice President?

There are n >= 4 students. The school has a Board of Directors, consisting of one president and three vice-presidents. The entire board consists of four distinct students. How can I prove that ...
1
vote
1answer
49 views

Product Of Series With Increment Powers

I found this interesting aptitude question and I don't know how to solve this genre of question. Any help is welcome :) $$\prod_{n=1}^{49}n^n=1¹\cdot 2²\cdot\ldots\cdot49^{49}=?$$ Thanks.
0
votes
2answers
47 views

Closed form formula for the given product

I'm working on a recurrence which give me the following solution: $$ f(n)=(2+1)(2+\tfrac12)(2+\tfrac13)\cdots\left(2+\tfrac1{\lg(n)}\right) $$ so for $n=16$, $f(n)$ is just like: $$ ...