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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26 views

Is it possible to extract the sum of products without having the individual values?

Consider two sequences A and B. They have the same size. I can calculate any individual statistics from either of them, like averages, products, summations, etc. However, because they are very large, ...
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11 views

Higher dimensions of the gauss integral to calculate product of determinate integral of sines

I have been looking at multiplying out the gaussian integral then converting it to spherical coordinates (or the higher order equivalent) $$\pi^{3/2}=\int_0^\infty\int_0^{2\pi}\int_0^\pi r^2e^{-r^2}\...
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23 views

Solve the following limit: $\lim_{n \rightarrow \infty}\prod_{1\leq k \leq n}(1+\frac{k}{n})^{\frac{1}{k}}$ [duplicate]

I came across the following problem while studying for the GRE. It states: Solve the following limit: $\lim_{n \rightarrow \infty}\prod_{1\leq k \leq n}(1+\frac{k}{n})^{\frac{1}{k}}$ (A). $e^{\frac{...
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2answers
50 views

Prove that the product of a family of topological vector space is Hausdorff space

I want to prove that: let $\Lambda \neq \varnothing$ and $E_{\alpha}$ topological vector space, for all $\alpha \in \Lambda$. Then $$E:=\prod_{\alpha \in \Lambda} E_{\alpha}$$ is a Hausdorff space if, ...
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15 views

Prove that $(\mu_1 \otimes \mu_2)\circ {\Pi_1}^{-1}=\mu_1$

Suppose $\Pi_1 :(\Omega_1 \times \Omega_2, \mathcal{F_1} \otimes \mathcal{F_2} ,\mu_1 \otimes \mu_2) \rightarrow (\Omega_1, \mathcal{F_1},(\mu_1 \otimes \mu_2) \circ {\Pi_1}^{-1})$ is a projection map ...
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9 views

Logarithmic production

Consider the product$$\prod_{n=2}^\infty\left(1+\left(\log_{\sqrt n^{\sqrt n}}t\right)^2\right)=\prod_{n=2}^{\infty}\left(1+\frac{(\log_nt^2)^2}n\right)$$the inner part goes to one for every given $t$,...
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20 views

Is there a standard terminology for the $A_c = \sqrt{A_1 \cdot A_2}$ where $A_1 = A_x(x,y)$ and $A_2 = A_y(x,y)$?

$A$ is a spatially variant function in 2D space. $A_1$ is a function calculated by varying at $x = x_1$, and $A_2$ is a function calculated by varying at $y = y_1$. Then, I want to estimate $A_c$ at $...
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28 views

Production expresion for Taylor series expansión.

i need help, I am doing the Taylor series development around the origin of: $\sqrt{z+i}$, i have considerated $\sqrt{z+i}=\sqrt{i} \sqrt{1+\frac{z}{i}}$ and make the variable change $x= \frac{z}{i}$ ...
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28 views

Phy product notation

I was looking at an answer about matrixes and I found this statement: $$ \pi_A(A)=\prod_{k=1}^n(A-\lambda_kI)=\prod_{k=1}^nS(D-\lambda_kI)S^{-1}=S\prod_{k=1}^n(D-\lambda_kI)S^{-1}=S\cdot 0\cdot S^{-1}...
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1answer
21 views

A summation involving the inverses of the products of the elements of subsets of a set

A big thank you in advance to all who have sacrificed their time to help me with the following problem. Consider a set $T$ containing the first $k$ natural numbers. First, we find all the $v$-...
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1answer
53 views

Unique solution of the equation $\prod_{i=1}^m(a_i+1)\prod_{i=1}^n(b_i)=\prod_{i=1}^m(a_i)\prod_{i=1}^n(b_i+1)$

Let $a_i$ be a sequence of $m$ distinct odd integers and $b_i$ a sequence of $n$ distinct odd integers. We have to prove that, $$\prod_{i=1}^m(a_i+1)\prod_{i=1}^n(b_i)=\prod_{i=1}^m(a_i)\prod_{i=1}^n(...
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1answer
42 views

How would I write the product of something whilst also omitting an element?

I have a function $$f(x_1,x_2,...,x_n) = \prod_{i=1}^{n}x_i^{\alpha_{i}}$$ I want to take the partial derivative $$\frac{\partial f}{\partial x_k}$$ Now I believe this will look like $${\alpha_kx^{\...
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1answer
57 views

closed form for $\prod_{k=0}^{n}\left(1-2\alpha \cos\frac{2\pi k}{n}\right) $

Does anybody know a closed form for this multiplication? $$\prod_{k=0}^{n}\left(1-2\alpha \cos\frac{2\pi k}{n}\right)$$ where $\alpha$ is a real number or maybe even a potential method to use to ...
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29 views

Calculate the product: $(\sin\frac{\pi}{12} + i\cos\frac{\pi}{12}) (\sin\frac{\pi}{6} + i\cos\frac{\pi}{6}) (\sin\frac{\pi}{4} + i\cos\frac{\pi}{4})$

Calculate the following product: $$ \left(\sin\frac{\pi}{12} + i\cos\frac{\pi}{12}\right) \left(\sin\frac{\pi}{6} + i\cos\frac{\pi}{6}\right) \left(\sin\frac{\pi}{4} + i\cos\frac{\pi}{4}\right) $$ ...
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21 views

How do I make a $\prod$ function explicit?

I've got a $\prod$ (product operator) function that I'm trying to make explicit. I've managed to convert everything else to explicit form, which we can call $g(x)$, except for this one part, so ...
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4answers
99 views

Does $\prod_{m=1}^\infty \frac{1}{m^2}$ have a closed form?

We know that $$\sum_{m=1}^\infty \frac{1}{m^2} = \frac{\pi^2}{6},$$ but what about the product of the reciprocal of the squares: $$\prod_{m=1}^\infty \frac{1}{m^2}?$$ Do we use a different product ...
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1answer
40 views

Any example of measurable spaces where the measurable rectangles form an algebra on the product space?

As far as I know, for any $(\Omega_{i},\sigma_{i})$ i=1,2, $\textit{A}=\{A_{1}\times A_{2}: A_{1}\in\sigma_{1}, A_{2}\in\sigma_{2}\} $ is not an algebra in general on $\Omega_{1}\times\Omega_{2}$ in ...
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1answer
52 views

Is $\prod\limits_{p}{p^\frac{1}{p}}$ convergent?

I am trying to prove or disprove if $\prod\limits_{p}{p^\frac{1}{p}}$ converges. I have tested up to 400K and got the following value: $$\prod_{p}{p^\frac{1}{p}}=0.26431187257195837519$$ while for ...
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1answer
35 views

Quotient of cartesian product by the right action of a group

I've been recently reading about Burnside rings and I found Serge Bouc's paper. In one of its sections he explains different kinds of functors that will be considered in further reasoning. I got stuck ...
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1answer
19 views

How to check if a number can be represented as product of 2 consecutice numbers?

How to check if a number can be represented as product of 2 consecutice numbers? Eg 56 can be represented since 56 = 7*8 72 can be represented since 72 = 8*9
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1answer
36 views

Is there a relationship between the standard vector cross product and the vector cap product?

I just finished reading through Introduction to Matrices and Vectors, International Student Edition, by Jacob T. Schwartz. In chapter 6 they proposed a definition I've never seen before, that seems ...
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2answers
53 views

Show that this inequality is true

Show that $\frac{2}{3} \cdot \frac{5}{6} \cdot \frac{8}{9} \cdot ... \cdot \frac{999998}{999999} > \frac{1}{100}$. I tried to take another multiplication $\frac{3}{5} \cdot \frac{6}{8} \cdot \frac{...
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0answers
59 views

Is the product rule necessary?

I‘m not too good with maths and this is probably wrong. I was learning calculus and when the product rule was explained the teacher said it cant be simply multiplied. But I noticed that $(f(x)*g(x))$...
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41 views

Why $\frac{\sin(x)}{x} = (1-\frac{x^2}{\pi^2})(1-\frac{x^2}{4\pi^2})…$ [duplicate]

If we allow some $g(x) = \frac{\sin(x)}{x}$ such that $g(0) = 1$ and also some $f(x)$ where: $$f(x) = \prod_{k=1}^{\infty} 1 - \frac{x^2}{\pi^2k^2}$$ Then it is said f and g are equal. But my ...
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25 views

How can I show that this product is equal to a product of Gamma functions?

$\prod_{n=0}^{x-1}\left( 1+\frac{a}{an+b}\right) = \frac{\Gamma\left(\frac{a}{b}\right)\Gamma\left(x+\frac{a+b}{b}\right)}{\Gamma\left(\frac{a+b}{b}\right)\Gamma\left(x+\frac{a}{b}\right)}$ I found ...
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1answer
21 views

Question on the product topology as the coarsest topology so that projections are continuous

Let $(X_{j},\tau_{j})$ be topological space for $j\in J$ where $J$ is some arbitrary index set. Define the base of the product topology $\tau$ as $$\mathcal{B}:=\{\times_{j\in J}A_{j}: A_{j}\in \...
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2answers
59 views

Most simple expression of $ \prod_{k=1}^{n}k^k$

I'm looking for the most simple form, perhaps without product symbol, of $\displaystyle \prod_{k=1}^{n}k^k$ for any positive number $n$. Maybe this is already the most simple form? What I've done so ...
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1answer
27 views

Prove the identity between vectors A, B, C

2(A.B)(A.C)=(A.A)(B.C)+A^2.BC A^2 is neither a dot or cross product it's used in optics, to calculate the aberrations of a non-axially symmetric system
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1answer
293 views

Integrating the product of lines.

The Question. Suppose we have $n$ linear functions $f_k$ defined on $[x_1,x_2]$. Let $f_k(x_1)=y_k$ and $f_k(x_2)=z_k$ denote the function values at the endpoints of the interval. We would like to ...
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129 views

Trying to find a closed form for certain class of infinite products

I am trying to find out where there are closed-form for products of the form $$ \prod_{n=0}^{\infty} (1+ f_n(x) ) $$ In particular, to make it simpler, Im trying to find out for $f_n (x) = -x^{2^n} ...
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1answer
32 views

Prove that if $A\times B$ is nonempty then there is a $C$ in $A\times B$ such the the intersection of $(\bigcup C)$ with $A\times B$ is empty.

Prove that if $A\times B$ is nonempty then there is a $C$ in $A\times B$ such the the intersection of $\bigl(\bigcup C\bigr)$ with $A\times B$ is empty. I've been struggling with this problem for ...
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1answer
29 views

Bounding the time to multiply the first $n$ primes as $\sum_{i=1}^n \log{i}\log\log{i}$

I'm trying to bound the number of operations for computing the product of the first $n$ primes: $$\prod_{i=1}^n p_i,\ \text{where}\ p_i\ \text{is the}\ i\text{th prime}$$ Since the $n$th prime is ...
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1answer
74 views

How to prove the following strange relation concerning fibonacci numbers

Is the following relation concerning fibonacci numbers, $F_n$ true? $$F_{2n-1}^n=2^{2n^2}\prod\limits_{r=1}^{n}\prod\limits_{s=1}^{n}\left(\cos^2\frac{r\pi}{2n+1}+\cos^2\frac{s\pi}{2n+1}\right)$$ I ...
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3answers
68 views

Proving that $ \prod\limits_{k=1}^{n}{\left(1+\frac{1}{k^{3}}\right)}<\mathrm{e} $

How would you prove that $$ \left(\forall n\in\mathbb{N}\right),\ \prod_{k=1}^{n}{\left(1+\frac{1}{k^{3}}\right)}<\mathrm{e} $$ Wolfram|Alpha says its limit would be exactly $ \frac{\cosh{\left(\...
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1answer
32 views

Proving $\frac{b-a}{b-x}\cdot (\frac{b-a}{k+1})^{k+1}\geq (\frac{b-a}{k})^{k+1}$

Prove that: $$ |\prod_{i=0}^n(x-x_i)|\leq\frac{n!}{4}(\frac{b-a}{n})^{n+1} $$ where $x_i=a+i\frac{b-a}{n}$ for $i=0,...,n$ and $x\in [a;b]$ I have tried to do induction: I proved it for $n=1$ and now ...
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2answers
45 views

What are the coefficients of an expanded polynomial with given roots

I want to find the coefficients $h$ of an $n^{th}$ order polynomial with given roots $a$. The $n^{th}$ order polynomial is given by the geometric series and summation: $$ \prod_{k=0}^{n-1} x+a_k = \...
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1answer
22 views

How do you put conditions on indices in a sum of product?

I would like to know how to interpret the following mathematical expression $$ \sum_{i=1}^k \prod_{j≠i}^k f(i,j) $$ What puts me in trouble is the “ ≠ ” sign of the product. For example, if i=1, what ...
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37 views

Up to isometry, how many product metrics are there?

Given two metric spaces $(X,d_X)$ and $(Y,d_Y)$, you can form a product metric on the space $X \times Y$ by letting your metric be the $p$-product-metric for any $p \in [1,\infty)$ or the sup-metric ...
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2answers
26 views

Computing a summation of products more efficiently via factorization

Let $P = \{p_1, \dots, p_n\}$ (may be a multiset). I wish to compute $\sum\limits_{S \subseteq P \\ |S| = k} \prod\limits_{i \in S} i \prod\limits_{i \not\in S} (1-i)$ in an efficient manner. The ...
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47 views

What value does $\prod_{n=0}^\infty 1+{1\over x^n}$ converge to?

I would like to find out the exact value (if it can be written down and generalized) for the formula in the title for any x. I have got aproximate values, but I would apreciate an exact value of a ...
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2answers
54 views

Prove sum-product identity

I have verified numerically the following identity, for several values of $m,n$: $$\sum_{i_1, \ldots, i_m = 1}^n \prod_{k = 1}^{m - 1} (\delta_{i_k, i_{k + 1}} a_k + (1 - \delta_{i_k, i_{k + 1}}) b_k)...
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1answer
29 views

Proving the equivalence for the map given with wedge product

Let $\alpha \in \Lambda^{p}L$, which is $p$-th power of $L$, where $L$ is linear space of dimension equal to $n$. Let us consider the following map $f_{\alpha} \colon L \rightarrow \Lambda^{p+1}L$ ...
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0answers
24 views

Sum of products proof

Let $$ l_i(x)=\prod_{j=0,\ i\neq j}^n \frac{x-x_j}{x_i-x_j} $$ where $x_0,...x_n\in \mathbb{R}$ and $\forall_{i,j}\ i\neq j \implies x_i\neq x_j$ Show that: $$ \forall_{x\in \mathbb {R}}\ \sum_{i=0}^...
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1answer
49 views

How to show that $a^1 \simeq a$ for all $C$-objects $a$ in a cartesian closed category? (only one direction of isomorphism proof is needed)

I have already proven that $a \times 1 \simeq 1 \times a \simeq a$ given a terminal object $1$ of a cartesian closed category $C$ (CCC). By CCC I mean that $C$ is finitely complete and has ...
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1answer
36 views

Determining the parity (even or odd) of pi notation

I am trying to disprove a conjecture, and I have gotten it such that the conjecture is only true if $$\prod_{i=1}^{g}{(\frac{j_i^{L_i+1}-1}{j_i-1})}$$ is singly even (of form $2m$ where $m$ is odd). ...
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2answers
39 views

Prove $\sum_{i=0}^n \prod_{j=0,\ i\neq j}^n \frac{x-x_j}{x_i-x_j}=1$

Let $$ l_i(x)=\prod_{j=0,\ i\neq j}^n \frac{x-x_j}{x_i-x_j} $$ where $x_0,...x_n\in \mathbb{R}$ and $\forall_{i,j}\ i\neq j \implies x_i\neq x_j$ Show that: $$ \forall_{x\in \mathbb {R}}\ \sum_{i=0}^...
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3answers
25 views

Getting the highest product from n terms

In other words, you're looking for a partition a+b+c+d of a given quantity x such that the product abcd of the nonzero parts is maximum. Is this interpretation right? I don't know how to phrase the ...
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2answers
74 views

Is the product of 2 square roots of negative numbers a negative or positive value???

I couldn't quite tell if the product of two square roots of negative numbers in ℂ (complex numbers) should be a positive real number or a negative one.. and this is because of the following two ...
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0answers
17 views

A funtion $F(p,Z)$ defined like : $ \prod_{k=1}^n (1 + pk)$

A funtion $F(p,Z)$ defined like : $$ \prod_{k=1}^n (1 + pk)$$ Where p = constant and $nk \leq Z$; I have tried to find $$ \prod_{k=1}^{\lfloor Z/p \rfloor} (1 + pk)$$ as i can tell that the highest ...
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1answer
17 views

Prove that the square of the product from $k=1$ to $k=n$ of $2k$ is $2^{2n}(n!)^2$ [closed]

Prove that $$\left(\prod_{k=1}^n 2k\right)^2=2^{2n}(n!)^2.$$

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