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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.

8
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1answer
48 views

If integration is a continuous analog of summation (Addition), what is the continuous analog of multiplication (Product)?

One definition of integration over a continuous interval [a,b] into n subintervals with equal width $\Delta x$, and from each interval choose a point $x_i^*$. Then the definite integral of $f(x)$ ...
1
vote
1answer
35 views

Efficient method for computing the product of the first 8 terms of a recursive sequence

The problem I am trying to solve is the following: Let $x_1=97,$ and for $n>1,$ define $x_n=\frac{n}{x_{n-1}}.$ Calculate $x_1x_2 \cdots x_8.$ I tried the painstaking fail safe method for the ...
0
votes
0answers
20 views

What is the limit of the series (summation) of the q-Pochhammer symbol or the ~q-Pochhammer symbol?

I am interested in knowing if the following series converges or not: \begin{equation} \sum_{n=1}^{\infty} \prod_{i=1}^n \left(1-e^{-\sqrt{i}} \right) \qquad Expression \; 1 \end{equation} If that is ...
0
votes
0answers
13 views

How to write a function in R that will obtain this product?

Kaplan-Meier estimator: I've just started using $R$, so I don't understand how to build this function $($it is modified Kaplan-Meier estimator$)$. It should be something like: ...
2
votes
2answers
31 views

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $

Prove that $ (\frac{\sum_{i=1}^n x_i}{n})^{\sum_{i=1}^n x_i} \le \prod_{i=1}^n {x_i}^{x_i}$ $, \forall x_i>0, n\ge1 $ (The second sum in the left-hand side of the inequality is an exponent) I've ...
1
vote
1answer
28 views

Is the following equation valid for a binomial coefficient?

From my notes on the binomial series I deduced a formula for the value of a binomial equation using product notation: $ {n \choose r} = {\frac {\prod_{\lambda=0}^r(n-\lambda)} {r!}}$ I believe this ...
-3
votes
1answer
63 views

Is there a formula for a summation divided by a product of its terms?

$$\frac{\sum_{i=1}^{n}x_{i}}{\prod_{i=1}^{n}x_{i}}= \frac{1}{x_{2}x_{3}x_{4}...}+\frac{1}{x_{1}x_{3}x_{4}}+\frac{1}{x_{1}x_{2}x_{4}}...$$ There is a very clear pattern that each consecutive result ...
1
vote
1answer
30 views

Inequality for finite product

I'm recently came across the following theorem: Let $f$ be a function in $C^{n+1}[a, b]$ and let $p$ be a polynomial of degree $\leq n$ that interpolates the function at $n+1$ distinct points $...
0
votes
1answer
37 views

$X$ is connected and separable. $X=Y\times Y$. Does $Y$ has to be also connected and separable?

$I$ is a finite set. It is not hard to see that, if $X=\prod_{i\in I}Y_i$ is separable, then $Y_i$ does not have to be separable. But for this special case such that $Y_i=Y_j \ \forall i,j\in I$, I ...
2
votes
1answer
43 views

Simplifying a product of a series

I tried simplifying the product $$\prod_{k=1}^{\infty}\left[1-x^k\right]$$ by factoring it into $$\prod_{k=1}^{\infty}\left[\left(1-x\right)\sum_{i=0}^{k-1}x^i\right].$$ I am not very experienced in ...
1
vote
2answers
47 views

Converting a Product to a Sum

How can I convert $$\prod_{i=1}^k\left[\sum_{j=0}^{i-1}x^j\right]$$ to a sum? I have been trying to solve this product by inductive reasoning but I figured it was too complex... Is there an agebraic/...
1
vote
0answers
38 views

Cross product in 4 dimensions [duplicate]

Cross product is defined in three dimensions the resulting vector have the same magnitude as the area of the parallelogram formed by the 2 multiplied vectors and its direction direction is ...
1
vote
1answer
37 views

The derivative of an integral of a product of functions.

I'm trying to comprehend the following result, which is required for fractional calculus: Let $w(x,y)$ and $f(z)$ be two real functions, such that they both vanish at a point $a$. Then the ...
1
vote
1answer
48 views

In the derivative of the product of two functions , why (dx)² is ignored?

I was digging deeply in the fundamentals of calculus, I found that in the famous '3Blue1Brown' channel, when demonstrating the process of finding the derivative of the product of two functions , (dx) ²...
0
votes
1answer
42 views

Fastest structured way to get max(abc) if a+b+c=30

What is the fastest and structured way to get maximum of abc if a+b+c=n, say n=30? a,b,c are non-negative and can be non-integer.
0
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0answers
42 views

Inverse function of a product space

I want to prove the continuity of a function $f: (X_1,\tau_1) \times (X_2,\tau_2) \rightarrow (X'_1,\tau'_1) \times (X'_2,\tau'_2)$ where $f(x,y) = (f_1(x),f_2(y))$ and my question is: What is $f^{-...
3
votes
2answers
171 views

Closed Form of the Real Portion of $f(n) = \prod_{m=2}^{n-1} e^{\pi i n/m}$

I am wondering if it is possible to express an equation in closed form. I currently have: $$f(n) = \prod_{m=2}^{n-1} e^{\pi i n/m}$$ Where $i$ is the $\sqrt{-1}$, which I know it commonly ...
1
vote
2answers
42 views

Prove that product of 3 numbers with a fixed sum is highest when they are equal

How to prove the fact, that product of any three numbers with a fixed sum is highest possible when they equal $sum/3$, meaning they are equal? Example: Let the "sum" be a constant $K$ and the "...
0
votes
6answers
56 views

Proving the highest product of two numbers with a fixed sum

How to prove that any two numbers with a fixed sum have the highest product when both equal half of the sum? Example: Given sum equals $10$. Why is then $5\times5$ more than $6\times4, 7\times 3, ...
1
vote
1answer
25 views

Meaning of index of a multiplication symbol in a Cartesian product

I've just started to read about Category Theory. More precisely nLab, opposite category. I'm trying to understand the expression: $$ \circ_{C^{op}} : C_{\mathrm{mor}}^{op} {}_{s^{op}}\times_{t^{...
4
votes
2answers
141 views

Sum of positive elements divided by their “weighted” product - inequality

I have following expression, $$ \frac{\sum_{i=1}^n x_i}{\prod_{i=1}^nx_i^{p_i}} $$ where $p_i$s satisfy $\sum p_i = 1$ and $p_i \in [0,1]$ and $x_i\geq0$, $\forall i \in 1\dots n$. I think that ...
1
vote
1answer
42 views

Representing a linear combination of matrices as some kind of inner product.

If I have a linear combination of two numbers $\lambda_1y_1+\lambda_2y_2$, I can represent it as $\mathbf y \cdot\vec \lambda$, i.e., an inner product of two vectors. If I have a linear combination ...
2
votes
1answer
48 views

How is it obvious that $\times : C \times C \to C$ is right adjoint to the diagonal functor?

This is from "Sheaves in Geometry & Logic". $\times : C \times C \to C$ is the cartesian product of two objects. So assume that finite products exist in $C$ the above is a functor. To say ...
4
votes
2answers
85 views

How does one evaluate the multiplication $f(2)\cdot f(3)\cdot f(4)\cdots f(15)$ by formulating?

Suppose $$f : \mathbb{Z}^+ \rightarrow \mathbb{R}$$ $$f(x) = 1-\dfrac{1}{x^2}$$ How does one evaluate the multiplication $\prod_{i=2}^{15} f(i)=f(2)\cdot f(3)\cdot f(4)\cdots f(15)$? Here I have ...
3
votes
2answers
191 views

A difficult infinite product

I would ask you for some hints for the calculation of the product $$\prod_{n=1}^\infty(1-e^{-18 n\pi}).$$ Is it possible to be approached without special functions?
3
votes
0answers
55 views

What are the prime divisors of $\det(A_n)$, where $A_n$ is the $n\times n$ matrix given by $(A_n)_{i,j}={n\choose|i-j|}$?

For $n\in\mathbb{Z}_{\ge 1}$, let $A_n$ be the $n\times n$ matrix given by $(A_n)_{i,j}={n\choose |i-j|}$. From this post it is clear that $$\det(A_n)=\prod_{k=0}^{n-1}\left[\left(\exp\left(\frac{2\pi ...
1
vote
1answer
45 views

Why is $ \prod_{n=0}^{N-1} u[x_n + \theta] - u[x_n-\theta] = u[\theta - \max(|x_n| )]$?

I'm self-studying math, and came across a problem: $$ \prod_{n=0}^{N-1} \left(u[x_n + \theta] - u[x_n-\theta]\right) = u[\theta - \max(|x_n| )] $$ where $u$ is a unit step function, $x_n$ are sample ...
1
vote
2answers
40 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 ...
1
vote
1answer
38 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 ...
1
vote
1answer
62 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{...
0
votes
0answers
39 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 ...
0
votes
1answer
54 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 $\mathbf{s}$ ...
1
vote
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 $\mathbb{R}^n$...
3
votes
2answers
62 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,...
0
votes
1answer
27 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 $[f\cap(A_1\times B_1)=g]\wedge [g\cap(A_0\times B_0)=f]$ ...
0
votes
1answer
50 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. $\...
1
vote
2answers
58 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 ...
2
votes
1answer
61 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-...
3
votes
0answers
334 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}) ) $ ...
4
votes
2answers
121 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 ...
0
votes
0answers
30 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
votes
1answer
104 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 $n= 2,3,4,5,6,7,8...$ ...
0
votes
1answer
38 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_{\...
0
votes
1answer
33 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 ...
1
vote
0answers
57 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 ...
1
vote
0answers
39 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(...
0
votes
3answers
63 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, ...
0
votes
2answers
34 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 ...
0
votes
2answers
37 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: $...
0
votes
0answers
64 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?