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

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11
votes
3answers
114 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
1
vote
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
votes
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 ...
0
votes
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 ...
1
vote
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 ...
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
votes
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 ...
1
vote
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 ...
1
vote
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: ...
2
votes
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 $$ ...
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
vote
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
votes
2answers
40 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
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
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
vote
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: $$ ...
0
votes
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? $$\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right)...\left(1-\frac{1}{n^2}\right)$$
1
vote
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
votes
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
votes
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?
1
vote
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 ...
28
votes
4answers
790 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
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 ...
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
1
vote
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 ...
0
votes
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 ...
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 $\sum_{j=1}^m a_j = 1$ ...
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, ...
5
votes
2answers
50 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: ...
24
votes
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 ...
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 ...
10
votes
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 ...
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
vote
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 ...
0
votes
1answer
46 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
vote
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
30 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 ...
0
votes
1answer
24 views

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

$(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ this exercise is taken from Apostol's Calculus I (page 45) and it's supposed to be proved by using the binomial ...
0
votes
1answer
18 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 ...
1
vote
1answer
39 views

Product of two vectors

Let $x, y \in \mathbb{R}^n$, when $x^T y = y^T x$ ?
0
votes
0answers
19 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
1
vote
1answer
21 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
votes
1answer
54 views

Solving a Pi product.

How the value of this $P_k$ is calculated from the first equation. Thank you. $$k \geq m$$ $$P_k=P_0\prod_{i=0}^{m-1}\frac{\alpha}{(i+1)\mu}\prod_{j=m}^{k-1}\frac{\alpha}{m\mu}$$ ...
0
votes
2answers
37 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. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
1
vote
2answers
42 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 ...
0
votes
0answers
44 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
votes
2answers
107 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 ...