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

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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 ...
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53 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 ...
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3answers
22 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 ...
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1answer
65 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 & ...
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21 views

Product notation $\prod$ when product does not commute

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 ...
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32 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 ...
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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? ...
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2answers
25 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 ...
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2answers
70 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 $b$ ...
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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 ...
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1answer
161 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 $(n+1)(n+2)...(n+10)$ ...
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2answers
62 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 ...
2
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1answer
65 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 ...
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1answer
41 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 ...
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2answers
465 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 ...
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1answer
22 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 ...
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1answer
19 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 ...
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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 ...
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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 ...
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3answers
37 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...
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1answer
33 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 ...
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1answer
31 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 ...
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2answers
35 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 ...
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1answer
43 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.
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2answers
44 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: $$ ...
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0answers
28 views

Sum of products of numbers in a list

For N numbers in a list, is there a formula to get the sum of products of the numbers in the list? Example: {3, 3, 2} = (3 * 3) + (3 * 2) + (3 * 2) Right now I'm ...
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1answer
96 views

Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
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59 views

being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$?

Let's say that I have a vector $\mathbf{w}$. How can I calculate the derivative in the following expression? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$ Update: found these ...
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27 views

being $\mathbf{a}$ and $\mathbf{b}$ two vectors with same length, how do I expand $(\mathbf{a}^T\mathbf{b})^2$?

Let's say that I have two vectors $\mathbf{a}$ and $\mathbf{b}$. Assuming that they have same length, their product $\mathbf{a}^T\mathbf{b}$ and its square $(\mathbf{a}^T\mathbf{b})^2$ are scalars. ...
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2answers
62 views

How does the max of $\prod_i a_i$ work?

Here are two succinct statements of the 'same' question: Statement 1: Take $a>0$ and $S \subseteq \mathbb{R}^N; S=\{(x_1,\dots,x_N)| \frac{1}{N}\sum_i x_i = a; x_i>0\}$. Define a 'product ...
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1answer
14 views

Could the multiplication of matrix X (with dimensions [d+1 x N]) and its transpose simplify to a matrix with [d+1 x d+1] dimensions?

In a machine learning course I'm taking, one of the lectures deals with matrix multiplication. Could anyone explain why the dot product of these two matrices would "shrink" to [d+1 x d+1] ...
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38 views

Values of $x$ for convergence

I was posed this problem, it took me a while to solve it – but, I did nevertheless. I shall pose it for all of you, too. In my opinion it is a great exercise. For what values of $x$ is the series ...
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1answer
46 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
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2answers
48 views

Proper way to express 0 in this case?

If 0=(x-a)(x-b)(x-c)...(x-x)..=0. So it's a product sum that we write with pi instead of sigma but how? There should be indexes but I'm not convinced that I understand what notation to use. $$\prod_{ ...
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4answers
200 views

Finding $\frac{\mathrm d}{\mathrm dx} x!$

I'm trying to differentiate $x!$ but I just can't seem to do it right. I define the function as follows: $$x! = \prod_{r = 0}^{x}(x-r) \quad,\quad x \in \mathbb N$$ I've tried attempted to try it by ...
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1answer
65 views

Simplifying a product written in Capital Pi Notation

I'm having some trouble figuring out how to simplify Capital Pi Notation. What I tried was to expand the multiplication with various n and tried to find a pattern. Could someone point me in the ...
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2answers
58 views

Show that $H_i=H_{n-i}$ and $\sum H_i=1$

We define $$H_i=\frac{1}{n}\frac{(-1)^{n-1}}{i!(n-1)!}\int_{0}^{n}\prod_{j=0,j\neq i}^{n}(x-j)dx$$ This is called the Newton-Cotes coefficient. Here is the exercise: First, convince yourself that ...
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1answer
22 views

Inequality involving products

One is given two intervals $I_{a-\epsilon,b+\epsilon}$, $I_{a,b}$ of $\mathbb{R}^n$, and is asked to show that $\lambda(I_{a-\epsilon,b+\epsilon}) - \lambda(I_{a,b}) \leq c\epsilon$ for some constant ...
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48 views

How to compute product integrals?

From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b ...
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1answer
22 views

Existence of 3 Matrices with given restrictions

Would it be possible to have 3 square matrices (preferably 2x2 or 3x3) $A$, $B$ and $C$ such that: $A\neq B \neq C$; The product $A\cdot B\cdot C$ equals the Identity Matrix; All 3 matrices are ...
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1answer
61 views

Do we need $\mu, \nu$ to be $\sigma$-finite to show $\int fg \ d(\mu \otimes \nu) = \int f \ d\mu \int g \ d\nu$?

The problem statement: Let $(X, \mathcal F, \mu), (Y, \mathcal G, \nu)$ be $\sigma$-finite and $f \in \mathcal L^1 (\mu), g \in \mathcal L^1 (\nu)$. Show that $fg \in \mathcal L^1 (\mu \otimes ...
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47 views

ZigZag product - A simpler definition?

I have been fiddling with the ZigZag product and constructing expanders for a while now. I was wondering if the following definition of a ZigZag product is the same as the original article: Lets ...
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33 views

Product of tuples vs cartesian product of set

If $\left ( X_{i} \right )_{1\leq i\leq n}$ is an ordered n-tuple of sets their Cartesian product is defined as: $$\prod_{i=1}^{n}X_{i}:=\left \{ (x_{i})_{1\leq i\leq n} :x_{i}\in (X_{i}) \; \text{ ...
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1answer
30 views

dot product of vectors with not orthogonal basis

The dot produt (inner product in the context of Euclidean space) of two vectors $\mathbf{a}=\left [ a_{1},a_{2},...,a_{n} \right]$ and $\mathbf{b}=\left [ b_{1},b_{2},...,b_{n} \right ]$ is defined ...
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1answer
61 views

Proof of an inequality involving $(N-1)!$

How is it possible to prove the following inequality? ...
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2answers
44 views

Finding (or rather expanding) the product $(5-xy)(3+xy)$

Given the product $(5-xy)(3+xy)$ I tried the following, As we know, $(x+a)(x+b)=x^2+(a+b)x+ab$ Here $x$ is $xy$. But $xy$ has two signs$-$ and $+$. How do I solve this.
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1answer
47 views

tuple of tuples notation

Is the following notation right for indicating a $\mathit{m}-$tuple of $\mathit{n_{j}}-$tuples (I mean that each tuple of the $\mathit{m}-$tuple has a different number of elements)? ...
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1answer
32 views

Product of Chebyshev polynomials of the second kind?

So Wikipedia has this formula for a product of two Chebyshev polynomials of the second kind evaluated at a fixed $x$ with different indices: $$ U_n(x)U_m(x)=\sum_{k=o}^{n}U_{m-n+2k}(x) $$ Which would ...
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1answer
83 views

If $a + b + c = 0$ prove that

If $a + b + c = 0$, prove that 1)$$ \sum_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 3 $$ 2)$$ \prod_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 1 $$ There is a solution that uses two cubic ...
2
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0answers
116 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...