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

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1answer
39 views

Convergence of a product

I want to show the following product converges for $x<e$ and diverges for $x \ge e$: $$\lim_{n\to\infty}\prod_{i=0}^{n-1}\left(x-\frac{xi}{n}\right).$$ To do this, I would need to show that the ...
1
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0answers
35 views

Simplifying products

Sorry for the very general title, but I don't even know how to name my question. I got a formula which is: $f(n)=\prod_{i = 0}^{\infty} ((n \; \mathrm{rem} \; p^{i + 1}) \; \mathrm{div} \; p^i + 1) ...
1
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1answer
49 views

how is a factorial fraction equal to the product notation

how is the $\prod$(2k-3) from k=2 to n equal to : ${(2n-3)!\over 2^{n-2}(n-2)!}$ where n $>=$ 2 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 ...
0
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1answer
40 views

Product of randomly drawn numbers

Here are two code line to run in R: prod(rnorm(100, mean=1, sd=0)) # (1) prod(rnorm(100, mean=1, sd=0.2)) # (2) $prod(..)$ returns the product of a sequence. The sequence it given by ...
6
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3answers
233 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
2
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2answers
52 views

Generalizing the Product Rule

How would I go about generalizing the product rule to the product of $n$ functions $\psi_1(x), \ \psi_2(x), ..., \ \psi_n(x)$? That is, I'm hoping to obtain an expression for $$ \frac{d}{dx} \prod_{j ...
2
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2answers
263 views

Product of two complementary error functions (erfc)

I believe that (i.e., it would be convenient if, and visually appears that) the product of the two complementary error functions: ...
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1answer
46 views

A closed form for $\sum_{i=1}^{n} \prod_{k=1}^{i+2} (3k+2)$

I need to calculate the following expression. Is there any explanation to convert this expression into normal expression without those letters for sum and the product? Just normal expression. $$ ...
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0answers
53 views

Bound on product of degrees

Is there any more or less sharp bound on the product of the out-degrees of vertices in a directed graph (except for the ones with no leaving edges)? The graph may have multiple edges between two ...
0
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1answer
94 views

Preserving finite coproducts

i want to prove the following statement: Given a bicartesian closed category $\Bbb{A}$ (thus we have exponentials, finite products and finite coproducts) then the functor $F:\Bbb{A}\rightarrow\Bbb{A}$ ...
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0answers
51 views

Proving the convergence of a product

I have become interested in taking the $n^{th}$ term of a series and evaluating a product whose $n^{th}$ term is $(1+a_n)$. After looking around I came across the following inequality: ...
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1answer
46 views

Trigonometry and complex numbers

Suppose $z_0=e^{i\theta_0}$ a complexe number as $\theta_0\in ]-\pi,\pi[ \setminus\{0\}$. For $n\in \mathbb{N}$, we pose $z_{n+1}=\dfrac{|z_n|+z_n}{2}$ and $z_n=r_ne^{i\theta_n}$ with ...
0
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2answers
43 views

Curiosity in a product

I have noticed the following curiosity for a product of integers. Given an ordered (decreasing) sequence of strictly positive integers $(a_i)_{i=1 \ldots n}$, that is to say, such that: $$\forall\ i\ ...
1
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1answer
18 views

Validity of an inequality

Is this relation true ? $\Pi_{i=1}^n v_n \le \left(\frac{\sum_{i=1}^{n} v_n}{n}\right)^n$ Thank you
1
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1answer
27 views

Is there a fast way to compute coefficient of some term of the product of some series'?

The example in wikipedia is $$A=1-3x+5x^2-7x^3+9x^4-11x^5+\cdots$$ $$B=2x+4x^3+6x^5+\cdots$$ $$AB=2x-6x^2+14x^3-26x^4+44x^5+\cdots$$ And the term $x^5$ is given by ...
1
vote
1answer
189 views

Conditional expected value of a product of two independent normal variables

I'm trying to work out the following conditional expectation: $E[\epsilon_t z_t|\epsilon_t + z_t = k, \epsilon_{t-1}, z_{t-1}, \epsilon_{t-2}, z_{t-2},...]$ where $k$ is known and $\epsilon_{t}$ and ...
1
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1answer
235 views

How would one discover this finite product identity?

I recently found the following finite product identity in a table of products: \begin{align} \prod_{k=0}^{n-1}\left[\sinh^2y+\sin^2\left(x+\frac{k\pi}{n}\right)\right]=2^{1-2n}(\cosh(2ny) ...
0
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1answer
65 views

$\textbf{C}$-Monoids and products

i have a question about $\textbf{C}$-Monoids. We can make a new category $\textbf{Mon(C)}$ from the category $\textbf{C}$, namely the category of all $\textbf{C}$-monoids. A $\textbf{C}$-monoid is a ...
5
votes
5answers
2k 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 ...
0
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1answer
532 views

Dot product of the column vectors from a matrix and their transposes through matrix multiplication

I have a matrix with data, every dataset is a column vector in my matrix. I want to know the dot product of the transpose of each column vector with the original column vector. If I transpose the ...
2
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0answers
60 views

Why does $\frac{d}{d\theta} \theta\prod_{i=1}^nx_i = \sum_{i=1}^nx_i$

Is this just the product rule? I have this in my notes but I didn't think anything of it and now I'm wondering how this happens? Edit: Im working with maximum likelihood estimation and in my notes I ...
1
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1answer
297 views

Summation of a product

I have basic doubt regarding summation over a product. The starting equation is $\sum_{z}\prod_{k=1}^{K}\pi_k^{z_k}f(x|\mu_k)^{z_k}$ and it is given that $\sum_{k}{z_k}=1$ How does it become ...
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1answer
71 views

Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
0
votes
1answer
37 views

Algebraic formula for co-products in the category of digraphs

I define a digraph as a set $V$ (vertexes) and a relation $E$ (edges) on $V$. Morphisms of digraph are functions which preserve $E$. So we have a category. It is easy to show that products of $n$ ...
0
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1answer
68 views

Canonical direct product (in a category)

In some categories there are more than one (isomorphic) direct products: For example in Set there are $A\times B$ and $B\times A$ products (as well as many others). But only one of these products ...
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1answer
69 views

What are canonical injections for co-products in the category Rel?

What are canonical injections for co-products in the category Rel?
1
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1answer
29 views

Products/limits for non-small indexed families of morphisms?

Can the strange requirement that direct products exist only for small indexing families be relaxed, saying that all products (or limits) exists but some are outside of our category (and possibly ...
3
votes
1answer
131 views

About binary relations under certain conditions and their composition

(I have edited it. The previous version was with errors.) Let $A$ be a set. Let $\pi_0$, $\pi_1$ be projections from $A\times A$. Let $F_0$, $F_1$, $G_0$, $G_1$ be binary relations on $A$. Let ...
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0answers
29 views

Product of numbers and gaussian function

Trying to approximate a gaussian function $g(x)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp{\left(-\frac{1}{2}\frac{(x-\mu)^2}{\sigma^2}\right)}$ with another function I found the product ...
4
votes
1answer
45 views

Finding the minimum value of this product

Is it true that $f(n) = \prod _{ i=1 }^{ n }{ (1-\frac { 1 }{ { 2 }^{ i } } ) } \ge \frac{1}{4} \quad \forall n$? I came up with this expression while trying to find an alternative way to solve a ...
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1answer
50 views

For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative?

Let $k>0$ be an integer. For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative? Since $13$ is prime, and for $\gcd(m,13)=1$, $P(2m)=P(2)=2^{-12}$ (can be shown by considering the ...
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0answers
77 views

Direct products in a partially ordered category

Consider a category, whose set of objects is a poset. Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an ...
4
votes
4answers
97 views

Trying to figure out why for $n>1$ it's true that $\prod_{j=n}^{\infty} (1-2^{-j+1}) \geq 1/4$.

I'm trying to figure out why for $n>1$ it's true that $\prod_{j=n}^{\infty} (1-2^{-j+1}) \geq 1/4$. Any hints/answers/tips are greatly appreciated.
2
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1answer
113 views

Why is this a differentiable structure on the product manifold?

Suppose $M$ en $N$ are differentiable manifolds with differentiable structures $\{(U_a,x_a)\}$ and $\{(V_b,x_b)\}$ resp. Consider $M\times N$ and the mappings $z_{ab}(p,q):=(x_a(p),y_b(q))$ with $p\in ...
2
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1answer
105 views

To What Extent Does the Cartesian Product for Algebraic Structures Generalize?

I admit this question is quite general. If we have a group (or perhaps some other algebraic structure) $G$, we can define the Cartesian product $G\times G$ of $G$ with itself. And then powers of $G$ ...
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1answer
397 views

A product of two sequentially compact metric spaces is compact. How to prove this explicitly?

We know that a product of two (or finitely many) compact topological spaces is compact. And we also know that in a metric space, compactness is equivalent to sequential compactness. So a product of ...
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2answers
1k views

Proving the AM:GM inequality

I am doing past exam papers preparing for the finals and I came across this questions about three times: Prove that: $$\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\geq \sqrt[n]{a_{1}.a_{2}...a_{n}}$$ ...
0
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1answer
26 views

Finding $n$ such that $\sum^{n}_{k=0} \frac{2}{p_k} = \left ( \prod^{n}_{j=0} p_j^{-1}\right) p_x$

Let $p_n$ denote the $n$th prime. Is it possible to find $n$ such that $$\sum^{n}_{k=0} \frac{2}{p_k} = \left ( \prod^{n}_{j=0} p_j^{-1}\right) p_x$$ any other way than calculating both the ...
1
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1answer
245 views

Proof involving gamma function, infinite product and Gauss

How can I rigorously and directly prove that $$\Gamma (z)=\lim_{n\rightarrow \infty }\frac{n!n^{z}}{z(z+1)\cdots(z+n)}$$
4
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1answer
150 views

Is my proof correct? (the product $\prod_{n=1}^\infty (1+\frac{z}{n} ) \mathrm{e}^{-\frac{z}{n}}$ converges absolutely and uniformly on compact sets.)

I want to prove that the product $$\prod_{n=1}^\infty \left(1+\frac{z}{n} \right) \mathrm{e}^{-\frac{z}{n}}$$ converges absolutely, and uniformly on compact subsets of $\mathbb C$: My book ...
13
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2answers
2k 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 ...
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3answers
136 views

Product representations of the factorial function?

Is this the only product representation of the factorial function? $$ {n!} =\prod_{k=1}^{n} k $$
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6answers
121 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 $3n-1$ ?
1
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1answer
109 views

What does $\displaystyle\prod_{n\geq 1} \frac{n-z}{n+z}$ converge to?

Does the infinite product $$\prod_{n\geq 1} \frac{n-z}{n+z}$$ converge, and if so to what? It seems that $$\lim_{n\rightarrow\infty}\frac{n-z}{n+z} = 1$$ so it is reasonable to think that the product ...
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1answer
155 views

Whats the diffrence between Products and Coproducts

So I just started in on Category theory (reading the quintessential text, "Categories for the Working Mathematician"), and I am trying to get my head around the difference between Products and ...
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1answer
384 views

How to prove that $\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}(1-\frac{x^2}{n^2})$ [duplicate]

How to prove that $$\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$$ I tried it with the Taylor series of $\sin(x)$ but I failed. Is there any help?
0
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1answer
55 views

Different direct product in a category and its full subcategory

A question related to Continuing direct product on a subcategory. Let $F$ is a full subcategory of a category $G$. I denote $\operatorname{Ob}X$ the set of objects of a category $X$. Is it possible ...
0
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1answer
23 views

Continuing direct product on a subcategory

Let $F$ is a full subcategory of a category $G$, both categories having binary direct product. Is it always true that there is such a binary direct product in $G$ that it is a continuation of a ...
3
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2answers
73 views

Passage not understood in a Physics formula

I stumbled upon the demonstration of the energy problem and saw something I don't understand. I thought mathematicians would be happier to solve his kind of problem $$ \int_a^b \vec F \cdot d \vec s ...
0
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2answers
111 views

How to go from a sum to a product and a product to a sum?

I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?