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

learn more… | top users | synonyms

0
votes
1answer
21 views

product of polynomial [on hold]

How can I calculate this? $$\prod_{i=1}^n \left(1+x_i\right) $$ Can I use vieta`s formulas?
0
votes
1answer
26 views

Value of finite product based on empty set

How does one evaluate the following product if the set S happens to be empty? \begin{aligned} f(n)= n \prod_{x \in S} \left(1-\frac{1}{x}\right) \end{aligned} Is the value simply n or is it ...
0
votes
0answers
27 views

Linearise product of two non-negative variables

Is there a trick to linearise the product of two non-negative (decision) variables in linear optimisation? Let $x_1$ and $x_2$ be these variables with $0 \leq x_1 \leq a$, $a \in \mathbb{R}_+$ and $...
-1
votes
0answers
8 views

Product of B-splines

It's my first post and i'm delighted to be among you... I'm reading at the moment a paper "Some Identities for Products and Degree Raising of Splines" from Knut Morken. enter image description here He ...
54
votes
0answers
2k views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty ...
1
vote
1answer
21 views

Understanding components of a vector

I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector $\vec A$ in the ...
0
votes
1answer
27 views

Is there anything known in general about upper and lower bounds for $\prod_{i\leq n\vee p_n>k}(p_i-k)$

I have no specific reason to ask this question other than seeing that it comes up quite often when I'm playing around with prime numbers. Let $$f(n,k)=\prod_{i \leq n\vee p_n>k}(p_i-k)$$ Where $p_i$...
1
vote
1answer
34 views

Source of faulty reasoning in expectation of product of random variables?

For iid $\xi_i>0$, with $\mathbb E[\xi_i]=1$, what is $\mathbb E[\prod_i^M\xi_i]$? Approach 1: $\mathbb E[\prod_i^M\xi_i]=\prod_i^M\mathbb E[\xi_i]=1$. There is another approach for $M\gg1$ with ...
7
votes
1answer
666 views

Product of Fibonacci numbers

I'm looking for the asymptotic approximation of the product of the first $n$ Fibonacci numbers. Does there exist a tight approximation for these kind of things?
4
votes
1answer
61 views

The limit of consecutive positive integers which are the product of n primes.

The maximum length of a string of consecutive primes is 2: that is, the primes 2, 3. This is easily proven, as no even number other than 2 is prime. In contrast, consider the set of numbers which ...
0
votes
1answer
53 views

Product as the sum of powers times a symmetric polynomial: What's the name of this property and what is it used for?

I noticed that the product of a group of positive integers $N$ with $n$ elements can be expressed as the sum of powers of the smallest member of $N$ times some (what I later found out be called) ...
0
votes
1answer
48 views

Proving ${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)$

Let $A=2^{2^{-x}}$ and $B=2^{2^{-x}+1}(1+2^{2^{-1}})(1+2^{2^{-2}})\cdots(1+2^{2^{-x+1}})$ Showing $x\ge2$ $${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)\tag1$$ ...
0
votes
1answer
23 views

If two infinite sets have the finite closed topology, then their product does not have the finite closed topology.

Let $X_1$ and $X_2$ be infinite sets and $T_1$ and $T_2$ be the finite-closed topology on $X_1$ and $X_2$, respectively. Show that the product topology, $T$, on $X_2 \times X_2$ is not the finite ...
1
vote
1answer
40 views

Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$

I have seen this identity on Wolfram mathworld and in a comment to another similar trigonometric proof: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I can't seem to find a ...
1
vote
1answer
35 views

Modified Sum of Products

A given number k is to be expressed as a sum of products of integers keeping in mind that the integers used in above process do not exceed their cumulative sum as 100. For e.g., k = 19 can be ...
7
votes
1answer
76 views

Summation is to integration, as the large product operator is to…? [duplicate]

The integral is defined many ways but one that I am aware of is the Riemann Integral(?) which looks sorta like $\sum^n_{i=0} f(a +i\frac {b-a}n)*\frac {b-a} n$. An interesting thought is "is there a ...
0
votes
1answer
20 views

Product of all Square Roots, taken only Decimal Digits

How and where could I compute the decimal reminder of a product of square roots times ten: $$Dr\left( \prod_{x=1}^{k}x^\frac{1}{2} \right) \times 10$$ Where $k$ is a power of $10$. I would like to ...
2
votes
1answer
56 views

How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...
0
votes
0answers
67 views

Is this a finite product describing the partial harmonic series sums?

http://mathworld.wolfram.com/EulerProduct.html In the second last line, it gives a product P(n). Is this supposed to be describing the finite terms of the harmonic series sum? I don't see how it does....
-1
votes
1answer
41 views

Prove $2^i\prod_{j=1}^{i}\sum_{k=1}^{2j-1}(-1)^{k-1}k^2=(2i)!$

Prove identities (1) $$2^i\prod_{j=1}^{i}\sum_{k=1}^{2j-1}(-1)^{k-1}k^2=(2i)!$$ (2) $$2^i\prod_{j=1}^{i}\sum_{k=1}^{2j}(-1)^{k-1}k^2=(2i+1)!$$ Is there another approach to prove (1)? $$2^i\...
1
vote
1answer
33 views

Why is $\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$?

I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130). $$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$ ...
1
vote
0answers
19 views

In which cases are the main diagonal elements of a product of positive definite matrices positive?

Let $A$ and $B$ be symmetric positive definite (pd) $n \times n$ matrices and $C = A \cdot B$. In which cases is then every $c_{ii}$, the $i$-th main diagonal elements of $C$, positive? When $A$ ...
0
votes
0answers
15 views

Doing a project for a CALC II class, and need help determining if we have sufficiently solved a product problem.

We finished our final in Calc II, and now we are doing math projects to pass time until the semester ends. https://docs.google.com/presentation/d/1bYMKfCqcc9zG32Zs_Wti2Fp9rXV7Nbt4QxR3Vnp33Dc As you ...
0
votes
1answer
28 views

Spectra of matrices with null product

If I consider two square matrices $A$ and $B$ such that $A B = B A =0$ and I know eigenvalues and eigenvectors of $A$, is it possible to get informations about the spectrum of $B$? In particular, I ...
2
votes
1answer
30 views

Weierstrass Approximation Theorem for a Product Space?

I am faced with the following problem: Let $X$ and $Y$ be compact Hausdorff spaces and $f$ belong to $C(X \times Y)$. Show that for each $\epsilon > 0$, there are functions $f_{1}, f_{2}, \cdots , ...
1
vote
0answers
27 views

which values of k satisfies special property to formulate L function

Consider $x*\prod_{a=1}^{n}(1-x^a)^k$ Famously for k=24 this product satisfies the condition to be an L-Function. More information can be found here My question is for what other values of k, such ...
1
vote
1answer
16 views

Euclidean algorithm for dividing two products.

Say I have numbers, $a$ and $b$ represented as two products $$a = \prod_{i=0}^{N_a} a_i \hspace{1cm} b = \prod_{i=0}^{N_b}b_i$$ I do know $\{a_k\}$ and $\{b_k\}$ but can not store $a$ or $b$ in a ...
0
votes
0answers
21 views

Minimize and maximize the sum of dot products at the same time

this is the problem. I have a set of numerical positive vectors of equal length. For each pair of vectors $(\mathrm{i}, \mathrm{j})$ I define the vector $\mathrm{ij}=\mathrm{i} - \mathrm{j}$. I also ...
2
votes
1answer
39 views

Why does the product of adjugates equal an adjugate of the product?

How can I show that $\mathrm{adj} (AB) = \mathrm{adj}(B)\ \mathrm{adj}(A)$? It is obvious if determinants are non-zero, but if any of the matrices are singular, I just don't get it. UPD. I've just ...
0
votes
2answers
37 views

Why is my answer incorrect for this differentiation question?

$$y = x* ((x^2+1)^{1/2})$$ I must find $$dy/dx$$ $$u = x, v = (x^2+1)^{1/2}$$ To do this I must use the product rule and the chain rule. To get dv/dx, $$(dv/dx) = (1/2)*(b)^{-1/2}*2x $$ $$(dv/dx) ...
2
votes
0answers
108 views

What notation would I use to differentiate between a cartesian product and a cotangent bundle of surfaces?

If the $S^1$ is defined by $x^2 + y^2 = r^2$ , $T^2 = S^1 \times S^1$ is defined by $\left(\sqrt{x^2 + y^2} -R\right)^2 + z^2 = r^2$ , $T^3=S^1\times S^1\times S^1$ is defined by $\left(\sqrt{\left(...
4
votes
2answers
83 views

Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$

I want to prove $$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$ if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$ I have no idea where to start, any advice would be greatly appreciated!
1
vote
1answer
24 views

Find when the product would be an integer

The problem: The sequence $\{a_n\}$ is defined recursively by $a_0=1,a_1=\sqrt[19]{2}$ and $a_n=a_{n-1}a_{n-2}^2$ for $n \geq 2$. What is the smallest positive integer $k$ such that the product $a_1 ...
1
vote
1answer
45 views

Equivalence of definition of product in a category

I was reading Mitchel book on categories and the following observation without proof is given: A family of morphisms given by $\lbrace p_{i}:A \to A_{i} \rbrace$ is the product of $A_{i}$ if and only ...
23
votes
7answers
761 views

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

Evaluate $$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\...
2
votes
2answers
78 views

Linear approximation to the product: $\prod_{k=0}^r\left(1+\frac12\left(\frac{\frac12+k+1}{\frac12+k}-\frac{\frac12+k}{\frac12+k+1}\right)\right)$

I have come upon with the next expression: \begin{equation} P_r=\prod_{k=0}^r \left(1+\frac{1}{2}\left(\frac{\frac{1}{2}+k+1}{\frac{1}{2}+k} -\frac{\frac{1}{2}+k}{\frac{1}{2}+k+1}\right)\right) \end{...
2
votes
1answer
90 views

Prove that $\prod\limits_{2 < p \leq y}\left(1-\frac{2}{p}\right)\sim\frac{D}{\log ^2 y}$ [duplicate]

I'm writing my bachelor thesis about Brun's sieve method and his theorem. In one proof I found this statement without further explanation. It is important to show that the product doesn't converge "...
-1
votes
2answers
30 views

By induction prove $\prod _{i=1}^{n} \frac{n+i}{2i-3} = 2^n (1 - 2n)$

I need to prove the following by induction. $\forall n \in \Bbb N$ $\prod _{i=1}^{n} \frac{n+i}{2i-3} = 2^n (1 - 2n)$ I know the steps to take but I'm failing to come to the right side of the ...
0
votes
0answers
11 views

Parentheses and Comma Notation

I came across the following formula for normalizing Smith-Waterman scores, and I do not understand what the SW(p1, p2) part is trying to notate. Does it perhaps refer to a product? Click here to see ...
0
votes
0answers
23 views

Product of directed partial orders

Is a product poset (with componentwise order) of nonempty posets a dcpo if and only if each multiplier is a dcpo? (for both binary and arbitrary products)
0
votes
0answers
25 views

Is there a constant $C$ such that $\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}\cdot C$?

By Mertens' third theorem: $$\prod_{p\leq x}\dfrac{p-1}{p}\sim\dfrac{e^{-\gamma}}{\log x}$$ But does there exist a constant $C$ such that: $$\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}...
0
votes
0answers
31 views

I am looking a comparison of this computation and Riemann's approach for $lcm(1,2\ldots,x)$

Looking a comparison with a reasoning due to Riemann, I ask to me about the behaviour as $x\to\infty$ of the following arithmetical function $$ \left( \prod_{n\leq x}n^{-\mu(n)}\right)\cdot \left( \...
9
votes
1answer
218 views

How many numbers $ N \le 10^{10}$ are the product of $3$ distinct primes?

How many numbers $ N \le10^{10}$ are the product of $3$ distinct primes? I can realistically calculate any $\pi(n), n < 10^{15} $ but I don't think it's possible to list all primes $>10^8$ in ...
0
votes
0answers
23 views

Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic to a ...
8
votes
2answers
453 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 $(n+1)(n+2)...(n+10)$ ...
0
votes
0answers
30 views

Formula for combinations involving product notation?

So after looking at the factorial formula and learning about product notation, I recognized this relation between them: $$\prod_{n=1}^kn=k!$$ And after fooling around and doing some trial and error, I ...
1
vote
1answer
38 views

“Binary-Like” Function?; In Consecutive Products as Multi-Factorials…

Summary Is there a function $Z(a,b)$ or how would one find such a function so that for $a,b\in \mathbb N$, it would produce $0$'s on for each $a$th step for each $b$th value? For example: $a=2$, ...
2
votes
1answer
122 views

how is a factorial fraction equal to the product notation

How is the $\prod_{k=2}^n(2k-3)={(2n-3)!\over 2^{n-2}(n-2)!}$, where $n \geq 2$ Note: I know that the $(2n-3)!$ is equal to the product of $2k-3$ from $k=2$ to $n$, but I can't figure out the bottom ...
2
votes
2answers
32 views

Gamma representation of certain sequence

I'm trying to find a gamma rep for $ 15 \cdot 13 \cdot 11 \cdot 9 \cdot 7 \cdot ... $ Steps so far: It's a simple sequence of $ n \cdot (n-2) \cdot (n-4) \cdot (n-6) \cdot (n-8)... $ and so on. ...
27
votes
3answers
6k views

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

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$