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

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2
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1answer
82 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
26 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
9 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
22 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 ...
0
votes
0answers
25 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( ...
1
vote
1answer
30 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 ...
9
votes
1answer
217 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
20 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 ...
8
votes
2answers
416 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
27 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
33 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
112 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
31 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. ...
25
votes
3answers
5k 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 $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
1
vote
0answers
14 views

Where can I find methods to evaluate products?

I found it was slightly difficult to find resources that discussed methods for evaluating products, like $\Pi_{n=0}^ka_n$ Preferably, I want to start with the basics and move through some readings on ...
1
vote
1answer
48 views

A property of product order

Let $\mathfrak{A}$ be a poset, let $a\in\mathfrak{A}$. By definition $$\star a = \{ x\in\mathfrak{A} \mid \text{there exists non-least } y\in\mathfrak{A} \text{ such that } y\le a \text{ and } y\le ...
1
vote
1answer
34 views

Complex inner product proof

I have just solved this problem in the real inner product space with $\langle \cdot , \cdot \rangle$ as the inner product. Show that in a real inner product space we have: $\langle x,y \rangle = ...
0
votes
1answer
30 views

Nice formula for a sum product

So suppose I have an ordered set of numbers: $(a_1, a_2, ..., a_n)$ and I want to express the following sum/product in an elegant manner: $ a_1 + a_1 a_2 + a_1 a_2 a_3 + ... + a_1 a_2 ... a_n $ I ...
0
votes
0answers
25 views

Resolving Zeros in Product of items in list.

Given the formula: $\sqrt [ 1/N ]{ \prod _{ n=1 }^{ N }{ { P }_{ n } } } $ where ${ P }_{ n }$ is a list of real numbers, e.g. [0.4, 0.3, 0.2, 0.1] And the ...
1
vote
1answer
43 views

Finding an upperbound for $\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$

I was wondering whether there exists a known upperbound for: $$f(n)=\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$$ For example: ...
1
vote
1answer
83 views

How many groups of order $2016$ exists, which are a direct product of smaller groups?

There are $6538$ groups of order $2016$ upto isomorphism. How many groups of order $2016$ are a direct product of (at least two) smaller groups ? I calculated an upper bound by summing the ...
1
vote
1answer
29 views

Closed formula for finite product series

I need to solve the recurrence: $$ \begin{align*} T(n) &= kT\left(\frac{n}{2}\right) + (k - 2)n^3 \\ &\textit{where}\; k \in \mathbb{Z}: k \geq 2 \\ &= ...
5
votes
1answer
55 views

Convergence of $\prod (1+ta_n)$ implies convergence of $\sum a_n$ and $\sum a_n^2$

Let $a_n$ be a sequence of real numbers and assume that $\prod _n(1+ta_n)$ converges for two non-zero values of $t$, say $t_1, t_2\in \mathbb R\setminus \{0, -1/a_1, \ldots, -1/a_i, \ldots \}$. ...
0
votes
1answer
28 views

Showing multiplication inequality using induction

Use induction to show that: $$\frac{1}{2}\frac{3}{4}\dots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n+1}} $$ for $n > 1$.
1
vote
1answer
42 views

Triples of natural numbers with same sum and product

Im looking at pairs of triples of natural numbers without repititions such that the sums of the two triples are equal and the products of the two triples are equal. To be precise: Let $x<y<z$ ...
3
votes
2answers
276 views

Is knowing the Sum and Product of k different natural numbers enough to find them?

Can we uniquely identify the set of k different natural numbers (no two are the same) by knowing only their sum and product (and the value of k itself)?
0
votes
1answer
31 views

Expressing a product in terms of the sum

While solving a problem, I got to the expression $$(-a+b+c)(a-b+c)(a+b-c).$$ I would like to express it in terms of the sum $a+b+c$. Is there any possibility?
22
votes
7answers
659 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)] ...
2
votes
2answers
80 views

Sum of all Products on Catalan numbers

how can I simplify this? let: $$ C_n = {{2n \choose n}\over n+1} $$ find: $$ \sum_{P_1 + P_2 + ... + P_k = r} \left(\prod_{j = 1}^k C_{P_j}\right) $$ thanks!
4
votes
5answers
1k views

The limit of infinite product

Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
5
votes
6answers
3k views

Can the limit of a product exist if neither of its factors exist?

Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists. Sorry if this seems elementary, I have just started my ...
3
votes
1answer
40 views

What is condition that the sum of $n$ complex numbers eaquals their product

Let $n\geq2$ and let $\{z_1,\dots,z_n\}$ be a set of complex numbers. Is there a condition on the $z_i$'s such that $$\sum_{i=1}^n z_i=\prod_{i=1}^n z_i$$ is identically true? For $n=2$ the ...
2
votes
1answer
33 views

How can I express such a product?

I know for example that $$\prod^{k}_{n=0} a_n = a_0 \cdot a_1 \cdot a_2 \cdot a_3 \cdots a_k$$ But what if I wanted to express $\space 3^k$ as a product? I know it sounds like a simple question, ...
1
vote
1answer
22 views

Limit of products in $x_n = 1-An^{-\alpha}$ and their summation

Suppose that we have $A >0, \alpha >0$, and for each $n$, define $x_n = 1-An^{-\alpha}$ such that for large $n$ we have $x_n \in (0,1)$. Also, define the product sequence, $y_n = \prod_{i=0}^n ...
6
votes
1answer
96 views

Simplify Product of sines

Is there a way simplify this product? $$ \sin\left({n} \frac{\pi}{2}\right) \sin\left({n} \frac{\pi}{3}\right) \sin\left({n} \frac{\pi}{4}\right) ...\sin\left({n} \frac{\pi}{n-1}\right) $$ And, is ...
3
votes
1answer
58 views

How can I prove that every finite product can be transformed to the given form?

Suppose, the permutations $a=(123)$ , $b=(12)(34)$ , $c=(12345)$ and $d=(12)(35)$ are given. I checked with GAP that the elemts $$a^jb^kc^ld^m$$ with $0\le j\le 2$ , $0\le k\le 1$ , $0\le l\le 4$ , ...
0
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0answers
27 views
0
votes
1answer
20 views

Elementary proof: division by integer makes real number smaller.

It's been so long since I took discrete math, maybe someone here can help me with what ought to be a straight forward and simple proof. Effectively I want to show this: Let a and b be positive ...
1
vote
2answers
35 views

cup product in relative cohomology; why the subsets $A$ and $B$ of $X$ have to be open?

I have a question about cup products in relative cohomology. In lecture we defined the cup product on singular cohomology as follows: Let $R$ be a commutative ring with unit $1_R$, let $X$ be a ...
5
votes
0answers
32 views

Prove $\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kx)\,dx$

Let $N>0$ and $a_0,a_1,...,a_N$ be any positive numbers. How to prove that $$\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = ...
0
votes
0answers
24 views

Approximate product by product

Let $\mathbb A _n = \{a_1, \ldots, a_n\} \subset \mathbb R_+$. For given $n, K \in \mathbb N$ can we bound from above the following: $$\left|\prod _{k=1}^K x_k - \prod _{\ell=1}^Lb_\ell \right| \leq ...
0
votes
2answers
58 views

Pseudocode of brute-force algorithm that finds largest product of two numbers in a list [closed]

This one will require a basic knowledge of some computer science concepts. I am trying to come up with a pseudocode brute-force algorithm that finds the largest product of two numbers in a list $a_1, ...
2
votes
2answers
72 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) ...
1
vote
4answers
152 views

Why is this equal to 1?

Why is $$ \prod_{i=4}^0 (4i -1) = 1 $$ At least according to: http://www.wolframalpha.com/input/?=prod_{i%3D4}^0+%284*i+-+1%29 It is rather unintuitive, why would the product even be defined? One ...
0
votes
1answer
28 views

Why the usage of $H$ in the dot product $x^H y$ instead of $T$ or $'$ for transpose?

Given two vectors $v, u \in \mathbb{R}^n$ (i.e. column vectors), then the dot product of them is defined like this $$\sum_{i=1}^n v_i * u_i$$ Or usually it can be expressed in matrix-product ...
2
votes
0answers
41 views

How can I find the elements generating a group in a special way?

Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le ...
1
vote
3answers
33 views

Prove that $\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - \sum_{n \in \mathbb{N}}{a_n}$

The following proof is obtained from this paper. My question is how to obtain the inequality. My guess is because of the following inequality: $$\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - ...
0
votes
1answer
38 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} } ...
0
votes
0answers
11 views

To clear for variable 'a' in a sum of dependent products

I can't seem to find a way to clear this equation for variable $a$: $E[k] = \displaystyle\sum_{k=1}^nk\frac{a}{n+a-k}\displaystyle\prod_{i=0}^{k-1}1-\frac{a}{n+a-i}$ Do you think it's possible? Any ...