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Questions tagged [products]

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

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9 views

when is product of rectangular functions not equal to zero given different centers?

I'm self-studying math, and came across a problem: $$ \prod_{n=0}^{N-1} u[x_n + a] - u[x_n-a] $$ where u is a unit step function, $x_n$ are sample points, $a$ is a constant. So basically the centers ...
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2answers
38 views

Is this an empty product?

Assume that we have: $$\Psi_{j,k}=(t_{j+1}-\tau)\times...\times(t_{j+k-1}-\tau)$$ that can be written as: $$\Pi_{i=j+1}^{j+k-1} (t_i-\tau)$$ If we have $k=1$ , my text-book says that this is an ...
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1answer
32 views

Multiplication of a vector by an orthogonal matrix

I have a question, consider $V$ an orthogonal matrix, and $u$ and $z$ are vectors, and W is a matrix does : $V'u = W V'z \implies u = W z$ ? I want to get rid of the orthogonal matrix $V'$, my ...
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1answer
58 views

Why is $\prod_{k=1}^m{(e^{ikx}+e^{-ikx})}=\sum_{\epsilon_{k}=\pm1}{e^{i(\epsilon_1+2\epsilon_2+\cdots+m\epsilon_m)x}}$

Problem A5 in the 1985 Putnam Competition: Let $I_m=\int_0^{2\pi}\cos(x)\cos(2x)\cdots\cos(mx)dx$. For which integers $m$, $1\leq m\leq10$, do we have $I_m\neq0$? The solution rewrites $\cos(x)=\frac{...
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2answers
68 views

How can I prove $ \lim_{n\rightarrow\infty} (\frac{1}{e^n}(\frac{(2n+1)^n}{\prod_{k=1}^{n}(2k-1)}))=\sqrt{\frac{e}{2}}$ [closed]

How can I prove that the limit $$ \lim_{n\rightarrow\infty} \left(\frac{1}{e^n}\left(\frac{(2n+1)^n}{\prod_{k=1}^{n}(2k-1)}\right)\right)=\sqrt{\frac{e}{2}} $$
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2answers
76 views

Showing that $\prod_{k=1} ^{\infty}\left(1- \frac{1}{2k}\right) = 0$ [closed]

How to find this product? $$\prod_{k=1} ^{\infty}\left(1- \frac{1}{2k}\right)$$ I know the answer is $0$, but I want to know how and why.
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0answers
38 views

Product of evenly spaced factorials

I am attempting to get a closed form expression for a product of factorials $$(a!)\cdot(2a)!\cdots (na)!$$ where $a$ is a positive integer. Mathematica seems able to compute this whichever $a$ I give ...
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1answer
44 views

Is there a name for this 'multiplication table' operation of vectors?

Is there a name for an operation or a transformation that generates a matrix from two vectors, creating a multiplication table from its elements? This might be like multiplying a vector $\mathbf{s}$ ...
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1answer
17 views

What does the term in the right mean?

Reading an article about clustering, I came across this equality: It's basically a sum over $ \mu_{i,j}(1-cos(x_i, p_j))$ (where $ \cos(x_i, p_j) $ is the cosinus between two vectors in $\mathbb{R}^n$...
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2answers
61 views

Why is $\alpha^n\beta^m+\alpha^m\beta^n\le\alpha^s\beta^t+\alpha^t\beta^s$ if $(n,m)\preceq(s,t)$?

I've recently come across the following statement: $$\alpha^n\beta^m+\alpha^m\beta^n\le\alpha^s\beta^t+\alpha^t\beta^s$$ for $\alpha,\beta\ge0$, $n,m,s,t\in\mathbb N_+$, $n+m=s+t$, and $(n,m)\preceq(s,...
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1answer
22 views

Comparing two binary relations using Cartesian products

Let $f$ be a binary relation between sets $A_0$ and $B_0$ and $g$ be a binary relation between sets $A_1$ and $B_1$. It can be proved that $[f\cap(A_1\times B_1)=g]\wedge [g\cap(A_0\times B_0)=f]$ ...
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1answer
45 views

Value of $\prod_{n=2}^{\infty} \frac{n^2+1}{n^2-1}$

Consider the following product: $$\prod_{n=2}^{\infty} \frac{n^2+1}{n^2-1} = \prod_{n=2}^{\infty} \frac{1+\frac{1}{n}}{1-\frac{1}{n}}\approx 3.67608...$$ It seems to be close to OEIS A156648, i.e. $\...
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2answers
48 views

When is $\prod_{n=2}^{k} n-\frac{1}{n}$ not an integer?

For which values of $k$ does the following product not evaluate to an integer: $$\prod_{n=2}^{k} n-\frac{1}{n}$$ I find it somewhat surprising that it not only can evaluate to an integer, but also ...
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1answer
57 views

Prove that $\sum_{{a_{n-1}}=1}^{a_n}\,\sum_{a_{n-2}=1}^{a_{n-1}}\,\ldots\,\sum_{a_{1}=1}^{a_2}\,a_1=\frac{\prod\limits_{i=0}^{n-1}\,(a_n+i)}{n!}$.

Use Principle of Mathematical Induction to show that, for every integer $n\ge2$, $$\sum_{{a_{n-1}}=1}^{a_n}\,\sum_{a_{n-2}=1}^{a_{n-1}}\,\ldots\,\sum_{a_{1}=1}^{a_2}\,a_1=\frac{\prod\limits_{i=0}^{n-...
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0answers
83 views

Why is $\inf g \sup g = \frac{9}{16} $?

Consider this question here : Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $? Call that conjecture about $\frac{5}{4} $ conjecture $1$. Let $g(n) = \prod_{i=0}^n (\sin^2(n) + \frac{9}{16}) ) $ ...
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2answers
103 views

sum of an infinite series $\sum_{k=1}^\infty \left( \prod_{m=1}^k\frac{1}{1+m\gamma}\right) $

I am trying to find a closed form expression of $$ \sum_{k=1}^\infty \left( \prod_{m=1}^k\frac{1}{1+m\gamma}\right) $$ where $\gamma>1$. I've been trying this for a long time. Is there an easy way ...
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0answers
26 views

Sum of products of polylogarithm function

I want to find a nice expression of sum of products of polylogarithm function \begin{align} \sum\limits_{k_1+k_2+\cdots+k_m=k\\ k_1,k_2,\ldots,k_m\ge 1}Li_{k_1}(x)Li_{k_2}(x)\cdots Li_{k_m}(x)=? \end{...
3
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1answer
102 views

How to expand product of $n$ factors.

I have a product say \begin{equation} F(a,n,x) = \prod _{j=0}^{n}(1-{a}^{n-2\,j}x) \end{equation} I want to expand and hope to have general terms of the coefficients. I did for $n= 2,3,4,5,6,7,8...$ ...
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1answer
37 views

Prove that $f(\alpha)=A_\alpha$ is injective.

Let $\{(X_\alpha,\mathscr T_{\alpha}):\alpha \in \Lambda\}$ be an indexed family of Hausdorff spaces such that each $X_\alpha$ has at least two points, and let $X=\Pi_{\alpha \in \Lambda} X_{\...
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1answer
31 views

Should dot product must be applied on values of same scale

I have points in $n$-dimensions. I want to find the points which lie on one side of the plane and other lies on the second side and i'm trying to do this with the help of dot-product. Suppose i ...
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0answers
52 views

Inductive proof of Cotes' Theorem

I am trying to prove the following: $$\sin(nx) = 2^{n-1}\prod_{i=0}^{n-1} \sin(x + i \cdot 180/n).$$ Other proofs use topics from complex analysis, but I am trying to prove this result using a ...
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0answers
37 views

Closed form for product over Gamma function

Is there a "closed form" (with which I mean an expression not involving an indexed sum or product) for any of these four products? $$\prod_{k=1}^{n} \Gamma(\frac{x}{k*2+1})$$ $$\prod_{k=1}^{n} \Gamma(...
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3answers
60 views

Show that $\prod_{k=1}^n (n+k) = 2^n \prod_{k=1}^n (2k-1)$ for all $n\in Z^+$

Show that $$\prod_{k=1}^n (n+k) = 2^n \prod_{k=1}^n (2k-1)$$ for all $n\in Z^+$. I think I'm supposed to use induction to prove this, but I can't seem to figure out how. Any help would be appreciated, ...
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2answers
33 views

Definition of cartesian product when the index set is finite

Let $\{A_j | j\in J\}$ be be a family of sets. By definition, $$\prod_{j\in J} A_j = \{f: J \to \cup_{j\in J} A_j | f(j) \in A_j \}.$$ However, when we are given two sets, say $G,H$, we are not ...
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2answers
36 views

Dot product between two vectors and its independence from reference frame

I found in the equation (2) here the dot product between two vectors and the author said that it is independent from the reference frame. What does it mean in this particular case? The notation is: $...
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0answers
55 views

Verify Bayes rule using the relative frequency definition

Is it possible to verify the product and the Bayes rule for the discrete-valued case using the relative frequency definition of the probability?
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0answers
75 views

Does $\prod_{k=3}^{\infty}\cos\frac{2\pi}{n!}$ have a closed-form solution?

How to compute this please? $$\prod_{k=3}^{\infty}\cos\frac{2\pi}{k!}$$ I just know it is convergent.
7
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1answer
87 views

Proof of a bizarre identity

Prove That $$ \prod_{r=1}^{n} (1-x^r) = 1 - \sum_{r=1}^{n} (1-x)\cdot(1-x^2)\cdot \ldots\cdot(1-x^{r-1})x^r $$ I found this identity through experimentation with Wolfram Alpha while trying ...
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0answers
48 views

How to prove sine product formula?

I am trying to prove this formula: $\prod_{k=1}^{n-1} \sin (\frac{\pi k}{2n}) = \frac{\sqrt{n}}{2^{n-1}}$ I've tried few approaches: Taking log on it and transforming product to sum of logs. ...
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0answers
79 views

Closed form for a product of sines

From this question Evaluation of a product of sines I allready know that: $$\prod_{k=1}^{n-1} \sin(\frac{k\pi}{n}) = \frac{n}{2^{n-1}}.$$ I am interested in a closed form for the following product: $...
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8answers
281 views

Show that $\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)\cdots\left(1+\frac{1}{n^3}\right) < 3$

I have this problem which says that for any positive integer $n$, $n \neq 0$ the following inequality is true: $$\left(1+\frac{1}{1^3}\right)\left(1+\frac{1}{2^3}\right)\left(1+\frac{1}{3^3}\right)\...
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1answer
69 views

What is this product equal to?

I recently came across the following product over prime numbers, but lost the source of the formula. Can someone enlighten me? The product, taken over all primes $p$, is: $$ \prod _{p}^{\infty} \frac{...
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1answer
26 views

Integral product of polynomials

Let $p$ and $q$ be two quadratic polynomials, given by $$ p(x)=c_1+c_2x+c_3x^2, \qquad q(x)=d_1+d_2x+d_3x^2 $$ Express the integral $J=\int_0^1 p(x)q(x)\,dx$ in the form $J=c^TGd$, where $G$ is a $...
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0answers
31 views

Sequence product

I'm trying to find a closed form expression for the following product: $f(k) = \prod \limits_{m=1 \\m \neq k}^{N} \frac{1}{1-(\frac{k}{m})^{\alpha/2}}$. This can be simplified or approximated into: ...
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0answers
17 views

Product over empty set

Assuming some kind of function $f(x)$, what is the value of $\prod\limits_{x \in \varnothing}f(x)$? https://www.wikiwand.com/en/Empty_product suggests that it is equal to $1$ but I wasn't sure if it ...
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3answers
74 views

Prove that $\prod\limits_{k=1}^\infty \left(1+\frac1{2^k}\right) \lt e ?$

How would you prove that $$\displaystyle \prod_{k=1}^\infty \left(1+\dfrac{1}{2^k}\right) \lt e ?$$ Wolfram|Alpha shows that the product evaluates to $2.384231 \dots$ but is there a nice way to write ...
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2answers
23 views

Cartesian product with three sets

I was given a question that asks "Let A={1,2} ,B={x,3,y},C={2,y}. Find A x B x C and C x B x C " I have an example that I could go by but i'm not sure what they were multiplying together to get it. If ...
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2answers
104 views

Is this sum constant for n?

Hi I can prove that this sum is constant in $n\in \mathbb{N}$. However my proof is very long (a few pages with probability involved). Does anyone see a simple proof. The sum in question is (a q-series ...
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0answers
12 views

If $f(t)=\Pi_{i=1}^{d-1}\left(1+tb_{i}\right)^{\alpha}$ is concave on $0<t<1$, then $\alpha (d-1) \leq 1$

Let's $$f(t)=\prod_{i=1}^{d-1}\left(1+tb_{i}\right)^{\alpha}$$ with $b_{i} > 0$. I want to show that if $f(t)$ is concave on $0<t<1$, then $\alpha (d-1) \leq 1$ I tried since $b_{i}&...
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1answer
19 views

Cartesian product of 2 dimensional

Let $R=\{(1,1),(2,2),(3,2),(4,1)\}$. Then how can I calculate $R\times R$?
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1answer
36 views

Is $NK=KN$ still even if only one of them is normal but both are still subgroups?

In this question Prove that the product $NK$ of two normal subgroups $N$ and $K$ of a group $G$ is a normal subgroup of $G$, and $NK=KN$., it is proved that $NK=KN$ if both $N$ and $K$ are normal ...
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1answer
102 views

How to calculate the following product?

How can we compute $$\prod_{i=n}^{1}\cos^{2}\left(\frac{2^{i}}{2^{n+1}}x\right)$$ for $0<x<\pi$ ? Attempt: $$ \begin{align*} \prod_{i=n}^{1}\cos^{2}\left(\frac{2^{i}}{2^{n+1}}x\right) & ...
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1answer
35 views

How to show that $\prod_{i=1}^n a_i = \prod_{i=1}^{r} a_i \times \prod_{i=r+1}^{n} a_i$?

I need to prove this multiplication property. Is it correct? Suppose that $1\le r\le n$. Show that: $$\prod_{i=1}^n a_i = \prod_{i=1}^{r} a_i \times \prod_{i=r+1}^{n} a_i. $$
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0answers
37 views

Extreme-mega-ultra-crazy hypergeometric functions

I've been lusting over hypergeometric functions, and came up with some questions. Here goes. I've defined the following functions, and I want to know if there are any closed forms for them. $$f_1(2;...
0
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0answers
26 views

Closed form formula for an expression involving sums of products.

I am stuck trying to simplify or find a closed form formula for the following expression: $$\sum_{i=0}^{n-1} (\prod_{y=i+1}^{n-1}\frac{y+1}{y+2} \div \prod_{x=i+1}^{k-2}\frac{x+1}{x+2})$$ The ...
1
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1answer
27 views

Product Notation Conventions

If a product has the following notation, does that mean the product has zero terms? If there is an expression after this symbol, is the entire quantity, with the product symbol included, equal to zero?...
3
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4answers
45 views

Equivalent Capital Pi Notation Expressions

So I was working on a probability question and then this expression came up. When I consulted the answers, I struggled to understand exactly how I would get from one expression to the other myself. ...
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1answer
47 views

$\prod_{i=1}^{n}(x+\alpha_i)$ series expansion

Let $I=\{1,2,...,n\}$ for any $n\in \Bbb N$. Assume that $\forall i\in I, \alpha_i\in\Bbb R$. What is the series expansion for $$p_n(x)=\prod_{i\in I}(x+\alpha_i)$$ I've noticed that $p_n(x)=xp_{n-1}(...
0
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2answers
27 views

Cardinality of product of subsets of a group

Let $A, B$ be finite subsets of a group $G$ (not necessarily finite). Is it true that $|AB| = |BA|$? More generally, is it true that $|ABC| = |ABC| =\cdots$ any permutation of three elements, if $C$ ...
1
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1answer
23 views

Is there a way to distribute a derivative into a product of arbitrary bounds like a summation?

What I am interested in is $$ \frac{d}{dx}\prod_{a=1}^{b}f_{a}(x). $$ I know that a derivative can easily be distributed into a summation, but what about an arbitrary product?