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

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2
votes
1answer
46 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
65 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 ...
-1
votes
1answer
37 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)? ...
1
vote
1answer
31 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
13 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
27 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
15 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
17 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
37 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
81 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 ...
4
votes
2answers
82 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
23 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 ...
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 ...
22
votes
7answers
716 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}[ ...
-1
votes
0answers
24 views

How can this be simplified further?

So I was doing some simplification on an expression, and I got stuck on this point, I couldn't find how to simplify it further. Here's the expression: $\dfrac{\sum\limits_{i=1}^{n} ...
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) ...
2
votes
1answer
88 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
10 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
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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
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
22 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
443 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
29 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
36 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
114 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. ...
26
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
15 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
36 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
26 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
88 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
32 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
58 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
29 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
48 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
278 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
32 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?
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 ...