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

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0answers
23 views

The product of distribution taken over Unions

I have a probability problem as follow: $\mathbb{P}\big[\mathop{\arg\sup}_{x \in \bigcup_{i\in \{1,2\}} \Phi_{k,i} } \mathcal{f}(x )\geq y \big] = 1-\mathop{\prod}_{x \in \bigcup_{i\in \{1,2\}} \Phi_{...
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2answers
339 views

How to do multiplication (capital pi) in WolframAlpha?

How do i ask this in WolframAlpha: $$\prod_{i=0}^{i=10} \sin{(i)}$$ I used $\text{product}(...)$ and $\text{multiply}(...)$ or even $\text{multiplication}(...)$ but they don't seem to work. I am ...
2
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1answer
170 views

Proving formulas with products of Fibonacci numbers

While digging through my old notes, I stumbled upon some formulas involving multiplication of Fibonacci numbers that I discovered about 7 years ago (being fascinated with Fibonacci numbers at the time)...
1
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1answer
49 views

What do you need to perform Karatsuba multiplication?

Karatsuba multiplication is usually defined in $\mathbb{N} \times \mathbb{N}$ and computes $$(aB^m+b)(cB^m+d)=acB^{2m} +[(a+b)(c+d)-ac-bd]B^m+bd$$ (where B is the base, usually 10) in only three ...
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3answers
57 views

About $0!=1$ and $a^0=1$ as cases of empty product.

Some useful ''conventions'' as $0!=1$ or $a^0=1$ are particular cases of an empty product, i.e. a product between elements of the empty set. I know that such product is defined as a convention by: $$ \...
0
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2answers
90 views

Simplifying the product $\prod\limits_{k=1}^n \left(1-\frac1{k^2}\right)$ [duplicate]

Can we simplify the given product to a general law? $$\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)...\left(1-\frac{1}{n^2}\right)$$
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4answers
90 views

Evaluate $\lim\limits_{n\to\infty}\prod\limits_{k=2}^{n}\frac{k^2+k-2}{k^2+k}$

I can't find the product of a sequence. We have $$\frac{(2+2)(2-1)}{2(2+1)}\frac{(3+2)(3-1)}{3(3+1)}...\frac{(k+2)(k-1)}{k(k+1)}$$ I am stuck with $$P=\frac{2(n+2)}{n^2(n-1)}$$ but that isn't ...
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0answers
51 views

Product of Polynomials of Binary Variables: Linearization

I have the following term (in the context of mathematical programming): $$\prod_{p = 1}^P [1 - z_p(1 - \lambda_p)]$$ where $\lambda_p \in [0,1]$ is a parameter and $z_p \in \{0,1\}$ is a binary ...
6
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2answers
185 views

Simplify $\prod_{k=1}^5\tan\frac{k\pi}{11}$ and $\sum_{k=1}^5\tan^2\frac{k\pi}{11}$

My question is: If $\tan\frac{\pi}{11}\cdot \tan\frac{2\pi}{11}\cdot \tan\frac{3\pi}{11}\cdot \tan\frac{4\pi}{11}\cdot \tan\frac{5\pi}{11} = X$ and $\tan^2\frac{\pi}{11}+\tan^2\frac{2\pi}{11}+\tan^2\...
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3answers
306 views

Find the product of a sequence

How to find the product of a sequence $$\frac{2-1}{2+1}{}\frac{3-1}{3+1}...\frac{n-1}{n+1}$$ The solution is $$P_{n}=\frac{2}{n(n+1)}$$ My question is can we approximate product with integration?
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1answer
29 views

Calculate new product value from known product, distance and shift

I have $p_1 = xy$ and the distance between $x$ and $y$ is $d = |x-y|$. I don’t know the values of $x$ and $y$ but I know the product and distance between them, I want to get new product $p_2$ after ...
32
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4answers
1k views

Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
2
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1answer
215 views

Distribution of the product of a Normal and an Exponential random variable

What is the probability distribution of $M$, given $M=V*X/k$, where $X$ is Normal, $V$ is Exponential, $k$ constant? Or, in the real world, the probability distribution of (Cost/k) where Cost=Price*...
2
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1answer
46 views

for which values of the pair of integers $(n,k)$ is $p(n,k) =1+\frac{2^{k}-1}n$ is prime?

let $p(n,k)= 1+\frac{2^{k}-1}{n}$ for a positive integer $n,k$ -for which values of the pair of integers $(n,k)$ : $p(n,k)$ is prime ? Any help is very welcom .Thank you
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1answer
80 views

On finite sums and products

I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for ...
0
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0answers
207 views

Sum of products of K numbers taken from N numbers in closed form

Let's say i have 5 numbers, $A,B,C,D,E$. I want to know the sum of all the possible products of some or all of these numbers each taken at most once. Instead of a lot of multiplications and additions ...
0
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0answers
81 views

Solving a ratio of summation

I have to solve the equation \begin{equation*} y = \frac{\sum_{j=1}^m a_j x'_j}{\sum_{j=1}^m a_j x_j} \end{equation*} We have $\sum_{j=1}^m a_j \frac{x'_j}{x_j} = 1$ and $\sum_{j=1}^m a_j = 1$ given....
6
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2answers
62 views

A trigonometric product

I have to prove: $$\prod_{i=1}^6 \left(2\cos\left(\frac{2^{i}\pi}{13}\right)-1\right)=1$$ I really have no idea about starting with this one. With the help of Wolfram Alpha, I noticed that: $$\...
25
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11answers
2k views

Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition? For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b \text{-...
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1answer
35 views

Link between two products

Could someone help me to solve this problem : Let's denote by $A_i$ the following product, $$ A_i = \prod_{\substack{k=1 \\ k\neq i}}^n (a_k - a_i) $$ Is there any link or simple formula between $...
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3answers
317 views

What is the probability that the product of $20$ random numbers between $1$ and $2$ is greater than $10000$?

Twenty random real numbers $a_1,a_2,\dots,a_{20}$ are chosen such that $1\le a_i \le 2$. What is the probability that their product is greater than $10000$? (By random, I mean each real number in the ...
4
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2answers
83 views

If $\prod\limits_{k=0}^5(5^{2^k}+6^{2^k})=6^x-5^y$, what is the value of $x-y$?

I think this might be a contest math question, so I'm tagging it as such. I don't know how to do something like this by hand (or if it's even possible, though I would presume it is if it's from a ...
0
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1answer
40 views

The product of multiple univariate Gaussians

What is the final result of $$I=\mathcal{N}_{x}(\mu_1,v_1)\,\mathcal{N}_{x}(\mu_2,v_2)\ldots\,\mathcal{N}_{x}(\mu_n,v_n)=\frac{1}{\sqrt{2\pi\,v_1} } e^{ -\frac{(x-\mu_1)^2}{2v_1} } \frac{1}{\sqrt{2\pi\...
1
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0answers
59 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to $\mathbb{N}$...
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1answer
147 views

Is the product of all objects of a finite category an initial object?

If the product of all objects in a finite category exists, is it an initial object? I presume so, but I'm still learning this subject and I can't make a proof go through. Advice welcome. (Not a ...
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1answer
58 views

Is the nth root of a product of n terms used in place of the average anywhere?

In applied usage we typically take the average of values or terms which is done by summing them and dividing by the number of terms (for simple average): $$\sum_{i=1}^n \frac{a_i}{n}$$ It dawned on ...
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1answer
76 views

Double Product of a series

So in this proof (please don't ask about it, it's not important and it would take ages to explain) there's this step where they "switch" the values of the series of the double products in the ...
1
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1answer
140 views

How to minimize $a \times b$ where $a^b≥x$?

For example, if $x$ is 1 billion, the smallest possible $a \times b$ will be $3 \times 19 = 57$. This is because: $2^{30} \ge 1000000000$ $2 \times 30 = 60 $ $3^{19} \ge 1000000000$ $3 \times 19 =...
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1answer
42 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at $...
4
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1answer
55 views

proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem

$$\left(1+\frac 1n\right)^{n} = 1 + \sum\limits_{k=1}^n \left\{\frac 1{k!}\prod_{r=0}^{k-1}\left(1-\frac rn\right)\right\}$$ this exercise is taken from Apostol's Calculus I (page 45) and it's ...
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1answer
25 views

Counting zeros in product of numbers

This is surprising a simple asked question... How many zeros does the product $25^5$,$150^4$ and $2008^3$ end with? (A)5 (B)9 (C)10 (D)12 (E)13 The problem is,I am not allowed to use calculator ...
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1answer
41 views

Product of two vectors

Let $x, y \in \mathbb{R}^n$, when $x^T y = y^T x$ ?
1
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1answer
31 views

Summing up decrementing geometric series?

Is there any easy way of summing up, $c,z \in R$ $z < 1, c < z $ $ k,n\in N$: $$\large\sum_{k=0}^{\lfloor\frac{z}{c}\rfloor}\prod_{n=0}^{k}(z-nc)^n$$ I'm searching for a formula to sum up ...
0
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1answer
60 views

Solving a Pi product.

How the value of this $P_k$ is calculated from the first equation. Thank you. $$k \geq m$$ $$P_k=P_0\prod_{i=0}^{m-1}\frac{\alpha}{(i+1)\mu}\prod_{j=m}^{k-1}\frac{\alpha}{m\mu}$$ $$P_k=\frac{P_0\left(...
0
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2answers
48 views

Prove $n!\mid\prod_{k=i}^{i+n-1}k$

I have no idea how to prove this, I haven't yet learned much about this kind of product. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
3
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2answers
151 views

Inequality and Induction: $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$ [duplicate]

I needed to prove that $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$, $\forall n \geq 1$ . I've atempted by induction. I proved the case for $n=1$ and assumed it holds ...
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2answers
79 views

non-cartesian set product?

Foremost, this question is asked from a point of a computer scientist undergrad, so please don't nag me for inconsistent notation, or lack of proper vocabulary. Is there a concept in mathematics for ...
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0answers
62 views

Product-rule for Jacobian calculation, i.e. $\frac{d}{dx}(Ay)$ where A is a matrix and y a vector and both depend on x

I'm trying to understand a paper in which the author constructs sensitivity matrices in the process of linearizing an equation. I figured that the sensitivity matrix has to be a Jacobian Matrix, ...
3
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2answers
124 views

Product of repeated cosec.

$$P = \prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$$ I realize that there must be some sort of trick in this. $$P = \csc^2(1)\csc^2(3).....\csc^2(89) = \frac{1}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$ I ...
0
votes
1answer
108 views

Infinite product converges to meromorphic function

How do you show that $\frac{1}{z}\prod_{n=1}^\infty \frac{n}{z+n}(\frac{n+1}{n})^z$ is meromorphic? Any hints would be helpful, I'm having trouble bounding the functions and their logarithms. This is ...
4
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1answer
195 views

Sum and Product Puzzle and Prime Factors

Suppose we have two number $X$ and $Y,$ such that $1 < X < Y < 100,$ and $X + Y ≤ 100.$ Sue is given $S = X + Y$ and Pete is given $P = XY.$ They then have the following conversation: ...
2
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1answer
122 views

Measurable functions on product space

Let $(\Omega, \mathcal{H}), (E, \mathcal{E})$ and $(F,\mathcal{F})$ be measurable spaces. Let $(E \times F, \mathcal{E} \otimes \mathcal{F})$ be a product space. Define the following three functions: ...
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2answers
62 views

how do I prove this matrix result? [closed]

How do I prove that if A and B are lower triangular matrices, then AB is also a lower triangular?
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1answer
70 views

Building matrix expressions for product of sum, isolating vector of constants

This identity to build the matrix expression for the expression below is pretty straightforward: $$ \left.\sum\limits_{j=1}^M \left( a_j \cdot f_{i,j} \right) \;\right|_{i=1}^N = \left[\begin{array}{}...
2
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1answer
46 views

Writing an expression as a product of products

I am currently dealing with the following expression: $$\left(\prod_{i=1}^{N-1}(\lambda_N-\lambda_i)\right)\left(\prod_{i=1}^{N-2}(\lambda_{N-1}-\lambda_i)\right)\cdots (\lambda_2-\lambda_1)$$ Is ...
0
votes
3answers
225 views

Fundamental theorem of algebra simple proof for rewriting with roots

My question is very basic, as I do not understand the concept of rewriting a (complex) polynomial into a product of terms using the roots of the polynomial. I have encountered the fundamental theorem ...
0
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1answer
79 views

How do I prove the following statement about the complement of a cartesian product?

How do I prove that this statement is true? $$(A\times B)^C=\left(A^C\times B\right)\cup\left(A\times B^C\right)\cup\left(A^C\times B^C\right)$$
2
votes
1answer
32 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
5
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0answers
163 views

Solving a question by using special products (Students debate to Teacher)

So today,we got back our exam papers,and we found a question marked wrongly and teacher said that it is wrong.We all students do NOT believe this.So here is what happened. Before reading the next ...
1
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0answers
45 views

Integral over a product

How the following integral can be computed: $I = \int_0^1 (x-a_1)^{b_1}(x-a_2)^{b_2}...(x-a_n)^{b_n} dx$? Here, $a_i,b_i$ are real numbers and $n$ is a natural number. Are there any techniques for ...