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

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3answers
27 views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
2
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1answer
21 views

independence of random objects when forming product spaces

Suppose we have two probability spaces $(\Omega_1, \mathscr{F}_1, \{\mathcal{F}^1_t\},\mathbb{P})$ and $(\Omega_2, \mathscr{F}_2, \{\mathcal{F}^2_t\},\mathbb{P}_2)$, if we take product space $$\Omega ...
3
votes
1answer
34 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...
4
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2answers
68 views

Product rule for Hessian matrix

Let $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$. Is there a general formula for the Hessian matrix of their product? That is, what is $H(f(x) g(x))$, where $H(f(x)) = ...
1
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2answers
93 views

Product limit with exponentials

Find an explicit formula for the limit: $$\lim_{n \rightarrow \infty} n \prod_{k=2}^{n} (2 - e ^ {\frac 1 k})$$ I am not asking for convergence proof since I know the sequence is decreasing and ...
0
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2answers
40 views

Proof of associativity of polynomials product (infinite variables)

The product of polynomials in $R[X_i]_{i\in I}$ where $I$ is not necessarily finite is associative ($R$ commutative ring), but I can't find any detailed proof of this fact. Either it is left in ...
28
votes
4answers
789 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 ...
15
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2answers
296 views

How to compute the following integral in $n$ variables?

How can the following integral be calculated: $$ I_n=\int_0^1\int_0^1\cdots\int_0^1\frac{\prod_{k=1}^{n}\left(\frac{1-x_k}{1+x_k}\right)}{1-\prod_{k=1}^{n}x_k}dx_1\cdots dx_{n-1}dx_n $$ There should ...
2
votes
3answers
473 views

Change from product to sum

We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : ...
2
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1answer
52 views

What is this (unusual) matrix/vector operation called?

A typographical error let to an unexpected (but, for me, potentially useful) result: $$ \left\{\begin{array} & a & b & c\\ d & e & f \\ g & h & i ...
2
votes
1answer
60 views

Show that $\prod _{i<j}(x_i-x_j)$ can be divided wihout remainder in $\prod_{i<j}(i-j)$ [duplicate]

Let $x_1,...,x_n$ be a natural numbers, show that $\prod _{i<j}(x_i-x_j)$ can be divided wihout remainder in $\prod_{i<j}(i-j)$ I know $\prod \left(x_i-x_j\right)$ is the result of ...
1
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4answers
69 views

Expansion of $x^n-y^n$

Studying polynomials I couldn't find a way to expand $x^n-y^n$ as a product of other polynomials. Now of course we know that $$x^4-y^4=(x^2+y^2)(x^2-y^2)=(x^2+y^2)(x+y)(x-y)$$ and I came up with this: ...
1
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0answers
25 views

Product of Dependent Bernoulli variables

Let $B_{i,n}$ with $i=1,...,n$ be the triangular Bernoulli array defined as $$ B_{i+1,n} = B_{i,n}\,R_{i+1,n}+\left(1-R_{i+1,n}\right)\,F_{i+1,n}, $$ where $R_{i,n}$ and $F_{i,n}$ are iid Bernoulli ...
2
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1answer
15 views

Product of Bernoulli variates

I am stuck with something that looks very simple but I am not able to find where I am wrong. Let $\xi_k$ with $k=1,...,n$ be $n$ iid Bernoulli random variables such that $$ ...
1
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0answers
19 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\}} ...
0
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1answer
81 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 ...
0
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2answers
39 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 ...
0
votes
1answer
33 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 ...
1
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1answer
132 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 ...
0
votes
1answer
34 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 ...
1
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3answers
47 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: $$ ...
4
votes
2answers
522 views

Closed form expression for the product $\prod\limits_{k=1}^{n}\left(1 - \frac{1}{ak}\right)$

A simple method for evaluating a product is term cancellation. For example, the product $$\begin{align*} \prod_{k=2}^{n}\left(1-\frac{1}{k}\right)&=\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)\\ ...
0
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2answers
53 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
3k views

Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
1
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4answers
69 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 ...
0
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0answers
29 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 ...
0
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3answers
57 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?
14
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2answers
491 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$ [duplicate]

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
21
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1answer
599 views

Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$

If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then $$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$ This identity follows from a probability calculation ...
1
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1answer
20 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 ...
24
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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 ...
2
votes
1answer
73 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 ...
1
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1answer
7k views

Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2} $ aka the square root of the dot product... ...
2
votes
1answer
37 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
4
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4answers
4k views

Product of two cyclic groups is cyclic iff their orders are co-prime

Say you have two groups $G = \langle g \rangle$ with order $n$ and $H = \langle h \rangle$ with order $m$. Then the product $G \times H$ is a cyclic group if and only if $\gcd(n,m)=1$. I can't seem ...
1
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1answer
31 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 ...
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0answers
36 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 ...
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2answers
182 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 ...
0
votes
0answers
43 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 $\sum_{j=1}^m a_j = 1$ ...
21
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3answers
4k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
0
votes
0answers
28 views

Optimizing the trace of a matrix product

I have a problem where I have a NxT matrix P (lets just assume full rank for now, where N>>T) and a TxN inclusion matrix S. Each column of S must contain exactly one 1 and the rest 0's i.e. 1_T*S = 1, ...
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2answers
48 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: ...
16
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3answers
5k views

Why is there no explicit formula for the factorial?

I am somewhat new to summations and products, but I know that the sum of the first positive n integers is given by: $$\sum_{k=1}^n k = \frac{n(n+1)}{2} = \frac{n^2+n}{2}$$ However, I know that no ...
1
vote
7answers
180 views

Why does $(-1) \times (-1)$ give +1?

Why is $(-1) \times (-1)=+1$ ? What is the intuitive concept ? My second question : How can I show that no triangular number can be of the form $3n-1$ ?
3
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3answers
2k views

About the factors of the product of prime numbers

If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using ...
5
votes
1answer
102 views

With $N$ a constant $>0$, show $\prod_{p<x}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{\substack{p<x \\ p \ \text{prime}}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this ...
4
votes
2answers
73 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 ...
1
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0answers
47 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 ...
5
votes
1answer
100 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 ...
1
vote
1answer
28 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 ...