Questions tagged [products]

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

Filter by
Sorted by
Tagged with
-1
votes
0answers
65 views

The Product of the Roots of the Minimal Polynomial Representation for Sin$\big(\frac{\pi}{x}\big)$ for Rational x, Closed Form Needed

Here is a function that I have had fun with: if you type in solve minimalpolynomial[sin[pi/x]]=0 into Wolfram Alpha and type in x as a number, otherwise it will not know how to interpret your input, ...
4
votes
1answer
62 views

Is my direct proof correct for $ \prod_{i=2}^{n} \left(1- \frac{1}{i^2}\right) =\frac {n+1}{2n} $ good/correct? [duplicate]

I need to show via direct proof that: $$ \prod_{i=2}^{n} \left (1- \frac{1}{i^2}\right) =\frac {n+1}{2n} $$ We first note that $$1-\frac{1}{i^2} = \frac{(i-1)(i+1)}{i\cdot i}.$$ Then \begin{align} \...
2
votes
2answers
36 views

Projection from n-fold cartesian product to coordinates indexed by a fiber.

Consider the example 1.3.2 (xi) of Emily Riehl (2016) Category theory in context: I am having trouble trying to understand the part where $M^f$ is described. As far as I have understood, $M^f:M\times\...
-2
votes
4answers
66 views

Determine which is larger $15*2^{11}*8^{37}$ or $10*4^4*3^{77}$

How would I determine which of $15*2^{11}*8^{37}$ or $10*4^4*3^{77}$, taking the log and rewriting it, I find it comes down to comparing 77 to $113*3/5$. How would you solve it?
-1
votes
1answer
45 views

can anyone help me about the 'cauchy product rule' [closed]

can anyone help me about the 'cauchy product rule' part Property -4: (Addition Theorem) from generating function, $$E_n(x+y) = \sum_{k=0}^{n} \binom{n}{k}E_k(x)y^{n-k}$$ Proof: - We know from the ...
1
vote
1answer
47 views

Showing $(n+m)^{\underline{k}}=\sum_{v=0}^\infty{k\choose v}\cdot{(m)^{\underline{k-v}}}\cdot(n)^{\underline{v}}$ for falling factorials

Falling and rising factorials are defined as $$ \color{blue}{n^{\underline{k}}}=\,\,\color{blue}{\prod_{j=0}^{k-1}(n-j)} \qquad\qquad \color{blue}{n^{\overline{k}}}=\,\,\color{blue}{\prod_{j=0}^{k-1}(...
-1
votes
0answers
21 views

Have this relationship a name or theorem? (∑ xi) + C ~= k.∑xi [closed]

Please, is there a name or theorem for this relationship? (∑ xi) + C ~= k.∑xi In other words, I am replacing one o more terms in a sum by a constant the multiplies the rest of the sum. or more simply ...
2
votes
0answers
36 views

Compute the following product involving powers of roots of unity

Consider that one has to compute the following: $$ \prod\limits_{j=0}^{N-1} (1-\alpha\:\omega^{-j})$$, where $0<\alpha<1$ is just a real constant, and $\omega = e^{i 2 \pi/N}$ is a principal $N$-...
0
votes
0answers
21 views

I believe I need to take a certain kind of product of sets or compute permutations / combinations, but I'm not sure which…

Consider the following definitions... a = (c, d) b = (e, f) c = (g, h) d = (i, j) e = (k, l) f = (m, n) I need to compute some kind of product / permutations / ...
-2
votes
0answers
54 views

Mathematical notation for Cartesian Product with conditions

I know this should be pretty easy, but I'm trying to add conditions to a cartesian product. If we have the sets L and S, where $$L = L_1, L_2 .... L_n$$ and $$S = S_1, S_2 .... S_m$$ with each element ...
5
votes
1answer
79 views

Proving if $a_{k}\ge a_{k-1}+1$ then $1+\frac{1}{a_{0}}(1+\frac{1}{a_{1}-a_{0}})…(1+\frac{1}{a_{n}-a_{0}})\le \prod_{k=0}^{n}(1+\frac{1}{a_{k}})$

I've worked on this problem with the sequence $a_{k}$ being the natural numbers, that is $a_{k}=a_{k-1}+1$ and $a_{0}=1$. Over the naturals, $\prod_{k=0}^{n}(1+\frac{1}{a_{k}})$ can be proven to be $n+...
0
votes
2answers
66 views

Uniqueness of product in the category of Sets

I understand why the Cartesian product $A \times B$ is the product of $A$ and $B$ in the category of sets. But is it the only product from the point of view of the category theory? Following the ...
0
votes
0answers
18 views

Is there a simplified version of this product

How would one simplify the product $\prod_{i=1}^{N} (a_{i} + 1) $ ? I know that three of the terms would be $ \prod_{i=1}^{N} a_{i} $, $\sum_{i=1}^{N} a_{i} $ , and $1$ . The other terms seem to ...
2
votes
2answers
101 views

$\prod_{k \in Q}{\frac{k^2-1}{k^2+1}}$ is rational number for $Q \subset N$

Let's consider some subset of natural numbers $Q$ and product: $$\prod_{k \in Q}{\frac{k^2-1}{k^2+1}}$$ There is well known theorem that if $Q$ is the set of prime numbers than the above product is ...
-5
votes
0answers
52 views

Computing $\sin 1^\circ\sin 2^\circ\sin 3^\circ\cdots\sin 178^\circ\sin 179^\circ$

Compute $$\sin 1^\circ\sin 2^\circ\sin 3^\circ\cdots\sin 178^\circ\sin 179^\circ$$
0
votes
1answer
36 views

$(X^Y)^Z\sim X^{Y\times Z}$

I'm trying to prove the equipotency of these two sets. I know I have to find an invertible function $f:(X^Y)^Z\rightarrow X^{Y\times Z}$ between the two sets. Which means I have to find a function $g:...
0
votes
0answers
20 views

Can this finite product of complex-exponential terms be simplified?

$\displaystyle\prod_{n=1}^{M} 1+ a_n \left(\exp\!\left(-2i\pi \displaystyle\frac{k \, b_n}{N}\right)-1 \right)$ where $k$ is a non-negative integer, all $b_n$ are positive integers, all $a_n$ are ...
0
votes
0answers
35 views

How to define a double productory

I have the following expression (which I find in the article, Eq. (136).) $$\delta=\prod_{\mu, \nu =0}^{1}\left[\left(a_{1} + \left(-1 \right)^\mu a_{2} + \left(-1\right)^{\nu}a_{3}\right)^2 - 1\right]...
1
vote
1answer
39 views

Cellular homology of $X\times S^n$

I want to compute the cellular homology for $X\times S^n$ where $X$ is a CW-complex. My current observation is that the $k-$th cellular group will be generated by cells $\{e^k_i\times e^0, e^{k-n}...
2
votes
0answers
44 views

Using Viete's Product show that $\frac{3}{\pi}= \frac{\sqrt{2+\sqrt{3}}}{2} * \frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}{2} * …$

Viete's product is $$ \frac{2}{\pi}=\frac{\sqrt{2}}{2} * \frac{\sqrt{2+\sqrt{2}}}{2} * \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}*... $$ Show that $$ \frac{3}{\pi} = \frac{\sqrt{2+\sqrt{3}}}{2} * \frac{\...
3
votes
3answers
72 views

Calculating $\prod_{x=1}^{\infty}\frac{x}{x+1}$

I am trying to find the value of: $$\prod_{x=1}^{\infty}\frac{x}{x+1}$$ It is equal to $\frac12\times\frac23\times\frac34\times\ldots$ and we can observe that all terms are cancelling out. So it ...
0
votes
1answer
62 views

Multiplicative version of “summation”

Repeated sum is denoted using $\sum$ and is called "summation." What is the name for the analogous process with multiplication, denoted $\prod$?
2
votes
1answer
57 views

Sum of product of products

I was working on a proof, and found that, for my proof to hold, the following must be true: Let $\Lambda_{i} > 0$ for $i \in \{0, 1, .. , k\}$, such that $\Lambda_{i} = \Lambda_{j} \iff i = j$. ...
1
vote
1answer
40 views

what is the name of a regular (column * column) product of a vector

Let $A = |a_{0},a_{1}, a_{2},... a_{n} |$ and $B = |b_{0},b_{1}, b_{2},... b_{n}|$. If we define a vector function $product$ such that $product(A,B)=|(a_{0}b_{0}) , (a_{1}b_{1}) , (a_{2}b_{2}),\dots, (...
1
vote
2answers
47 views

Simplify $bx(x-b) + ax(a-x) + ab(b-a)$ to $-(a-b)(a-x)(b-x)$

I was doing some matrices problems when I got to this factoring $$bx(x-b) + ax(a-x) + ab(b-a)$$ I found this was the answer, but it was $$-(a-b)(a-x)(b-x)$$ Doing some researches, I found this ...
0
votes
0answers
40 views

Product of covariance matrices

I've a problem about the product of three covariance matrices. Let $\textbf{x}$ be a real random vector of dimension ${n}$, $\textbf{Y}$ be a real random matrix of dimension ${t} \times {n}$. Let $\...
1
vote
1answer
22 views

What is the difference between product of two functions and dot product of functions?

Given two functions e.g.: $$f(x)=\sin(x) \text{, } g(x)=2\cos(x)$$ Why can't we write: $$f\cdot g=\sin(2x)$$ But we use this: $$(f,g)=\int_{\alpha}^{\beta}f(x)g(x)dx$$ I don't quite understand if ...
2
votes
2answers
84 views

Evaluating $\prod_{k=1}^m \tan \frac{k\pi}{2m+1}$ [duplicate]

How to evaluate this? $$\prod_{k=1}^m \tan \frac{k\pi}{2m+1}$$ My work I couldn't figure out a method to solve this product. I thought that this identity could help. $$\frac{e^{i\theta}-1}{e^{i\theta}...
0
votes
0answers
29 views

If $a_1, \dots, a_n$ are all positive, show that $\prod_{k=1}^{n}(1+a_k) \leq \sum_{k=1}^{n+1}\frac{S^{k-1}}{(k-1)!}$, where $S = \sum_{k=1}^{n}a_k$ [duplicate]

I am having difficulties in providing the following proof: If $a_1, \dots, a_n$ are all positive, show that: $$ \prod_{k=1}^{n}(1+a_k) \leq \sum_{k=1}^{n+1}\frac{S^{k-1}}{(k-1)!} $$ where $S = \sum_{k=...
0
votes
2answers
37 views

Does there exist two infinite subsets of naturals $A, B$ such that each $n \in A + B$ uniquely determines $a \in A, b \in B$ such that $a + b = n$?

Does there exist two infinite sets of naturals $A, B$ such that each $n \in A + B = \{ a + b : a\in A, b\in B\}$ has a unique solution $a + b = n$ in which $a \in A, b \in B$? For example: $$ A = \{0,...
0
votes
1answer
61 views

What does $\prod_{p}$ mean without the upper value?

I was reading this paper and on page 2, I saw this: $$\prod_{p}\Bigl(\frac{1}{1-p^{-z}}\Bigr)$$ I know how to use a Pi when there is a value on the top, but what does it mean when there is no value on ...
1
vote
0answers
30 views

Showing that $G\left( \frac{\partial}{\partial b}\right) F(b) = F\left( \frac{\partial}{\partial x}\right)\left[G(x) e^{xb} \right]\Big|_{x=0}$

We want to show that \begin{align*} G\left( \frac{\partial}{\partial b}\right) F(b) = F\left( \frac{\partial}{\partial x}\right)\left[G(x) e^{xb} \right]\Big|_{x=0} \tag{1} \end{align*} By assuming ...
1
vote
1answer
58 views

Is there a name for this problem or any close problem?

I'm trying to figure out if there is a common name for the following optimization problem of computing products of subsequences of a sequence: Given a set $S$ of $m$ sets $\{S_1,\dots, S_m\}$, where $...
0
votes
1answer
26 views

Arranging a partial product

Why is it that for real numbers $0<x_i<1$ $(1-x_1)(1-x_2)\cdots(1-x_n)(1-x_{n+1})=1-x_1-x_2-\cdots-x_n-x_{n+1}+x_{n+1}(x_1+\cdots+x_n)$? The most I was able to do was factor out the $x$ terms ...
4
votes
1answer
49 views

prove :nth root of the Product of arithmetic sequence is bigger than the sqrt of the first and last numbers

let $ \{a_n\} $ be an arithmetic sequence of positive numbers. prove that for all $ n\ge2 $: $ \frac{a_{1}+a_{n}}{2}\ge\sqrt[n]{a_{1}a_{2}a_{3}\dots a_{n}}\ge\sqrt{a_{1}a_{n}} $ I managed to solve the ...
0
votes
1answer
17 views

Zero-ary cartesian product

Let $A=(A_1,...,A_n)$ be a vector of sets. The Cartesian product in $A$ could be defined as: $\times_A :=\left\{(a_1,..,a_n)\mid \forall a_i \in A_i\right\}$ Which one is correct? a) $\times_{\...
0
votes
1answer
29 views

Is there a series expansion formula for $\prod_{j=1}^{n} (a_j - b_j)$?

I would like to expand the product $\prod_{j=1}^{n} (a_j - b_j)$ into a summation. Since there is some $n$ number of factors, I cannot simply 'foil' out the expansion. Checking for any specific value ...
0
votes
1answer
22 views

Inequality involving product of iid random variables

Let $a>0$, $b>1$. Let $(X_k)$ be a sequence of iid random variables, with density $f(t) =\frac{a}{b} \left(\frac{b}{t}\right)^{a+1}$ for $t>b$ and 0 otherwise. I can work out that the CDF is $...
0
votes
0answers
46 views

Solving a strange type of equation, expressed in terms of scalar products.

I'm trying to solve the equation $$ -f(a+b,c,d)e^{a\cdot b}+f(a,b+c,d)e^{b\cdot c}-f(a,b,c+d)e^{c\cdot d}-f(a+d,b,c)e^{-a\cdot d}=0, $$ with $a,b\in \mathbb{R}^2$ and the scalar product $a\cdot b=...
1
vote
1answer
46 views

$\frac{1}{365^n}$ when all of the $x_i \in \{1,2,3,\ldots,365\}$ adds up to 1

Question: Let a discrete r.v be denoted $X_1,..,X_n$ denote the birthdays of $n$ people in a room. Assume that $X_1,..,X_n$ are mutually independent and that $X_i$ is a distribution such that $X_i \...
2
votes
1answer
55 views

Is how I evaluated the following product correct $\prod_{n=1}^{20}(1+\frac{2n+1}{n^2})$?

The product is $$\prod_{n=1}^{20}\left(1+\frac{2n+1}{n^2}\right)$$. I can rewrite this product as $$\prod_{n=1}^{20}\left(\frac{n^2+2n+1}{n^2}\right)$$ which can be further simplified to $$\prod_{n=1}^...
4
votes
0answers
67 views

An infinite product function

Before I start, must be said that I am not a math wizard, nor a math student. I just love nudging around with math, and I came across a random function I thought in my head (I do not take credit for ...
0
votes
1answer
31 views

Evaluating a long product using rules of product (Big PI)

$$T= \prod_{i=2}^{N} \left ( i^3 - 1 \right ) ~~ \text{Find T in terms of N}$$ This seems like a super easy task but somehow I cannot figure it out. My try was to convert it to a sum of logs, denote $...
2
votes
2answers
126 views

Proving that $f$ verifies $f(1−x) + f(x) = 1$

I have a function which is defined for $x \in (0,1)$ and for $p>1$ with the expression \begin{align*} f(x) = \sum_{k=0}^{p-1} \frac{(-1)^{p+k}}{(p-1-k)!(p+k)!}\left(\prod_{i=k-p+1, i\neq0}^{k+p} (...
2
votes
1answer
59 views

Proving $ \prod_{k=0}^{n-1} (n^2-k^2) = \frac{(2n)!}{2}$

I'm probably having a brain-fart but I can't figure out why this identity holds: $$ \prod_{k=0}^{n-1} (n^2-k^2) = \frac{(2n)!}{2} $$ I tried using various formulae involving $\binom{2n}{n}$, without ...
0
votes
1answer
29 views

Random Variable Multiplied with Independent Random Vector - Derivation of Product Distribution

Let $x$ be a random variable and $\mathbf{Y}$ be a random vector. Is there a general formula to derive the probability density function of $$ \mathbf{Z} = \frac{1}{\sqrt{x}} \mathbf{Y} \; ? $$ I could ...
1
vote
1answer
34 views

Efficient way of polynomial multiplication with dependency in Bipartite graph form

We use a Bipartite graph: $G=(U,V,E)$, with vertices: $U $ and $V$ to represent some task dependency. Every $U$ is a unique task, and every $V$ is a degree one polynomial, e.g. $(1+v_ix)$. Our goal is ...
0
votes
1answer
33 views

Product expression for roots of polynomial

I am working slowly through "Riemann's zeta function", HM Edwards, Dover Publications, 1974. At the top of page 18, I read (I have altered the notation from $p(s)$ to $P(x)$) "any ...
0
votes
1answer
43 views

Is it always possible to construct this integer sequence?

Let $A_n$ be a sequence of monotonically increasing, positive integers. How can one construct another sequence $a_n$, again positive integers, such that $$\lim_{n \to \infty} \frac{A_n}{\displaystyle \...
7
votes
1answer
126 views

An infinite product for $\frac{\pi}{2}$

Please help prove $$ \begin{align} \frac{\pi}{2}&=\left(\frac{1}{2}\right)^{2/1}\left(\frac{2^{2}}{1^{1}}\right)^{4/(1\cdot 3)}\left(\frac{1}{4}\right)^{2/3}\left(\frac{2^{2}\cdot4^{4}}{1^{1}\...

1
2 3 4 5
33