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

learn more… | top users | synonyms

5
votes
5answers
98 views

Showing that $\lim_{n\to\infty}\frac{1\cdot 3\cdot 5\cdots (2n-1)}{(2n)^n} = 0$

Ok, so I want to show that $$\lim_{n\to\infty}\frac{1\cdot 3\cdot 5\cdots (2n-1)}{(2n)^n} = 0.$$ Here is what I have tried so far: \begin{align} \notag \lim_{n\to\infty}\frac{1\cdot 3\cdot 5\cdot ...
2
votes
3answers
41 views

Sum and product of integers conserving order

I have the feeling this is true, but can't prove it: $$\sum_n^An\lt\sum_n^Bn\implies\prod_n^An\lt\prod_n^Bn$$ Where $A\subset\mathbb N-\{0, 1\},B\subset\mathbb N-\{0, 1\}$ Example: ...
0
votes
1answer
46 views

What's the maths symbol for alternation of product and sums?

Is there a mathematics symbol for referring to the equation below? (((((((((((100*y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x = 0, x = 9.8 I've tried using capital ...
1
vote
4answers
89 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
2answers
34 views

How to repesent n x m multiplication into symbol notation?

I am not a mathematician and so I might not be using the right terms. I have a vector of n components and another vector of m components ...
4
votes
2answers
95 views

A curious product formula

Fiddling with Mathematica seems to suggest the following: $$\frac{(2^2)(4^2)(6^2)\cdots(2N^2)}{(1^2)(3^2)(5^2)\cdots(2N-1)^2}=N\pi+\frac{\pi}{4}+\frac{\pi}{32N}-\frac{\pi}{128N^2}+o(1/N^2).$$ Does ...
0
votes
1answer
45 views

Proof of “factorization of polynomials” using only Complex Analysis

I ask for the proof of the following: If $p$ is a polynomial with degree $n\ge 1$ and zeros in $A\subseteq \mathbb C$ whose order (multiplicity) is given by $n:A\to \mathbb N^*$ then $A$ is finite ...
0
votes
0answers
87 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
2
votes
1answer
84 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
1
vote
1answer
43 views

Is it generally preferred that empty products are gotten rid of where possible?

Is it generally preferred that empty products are gotten rid of where possible? For example: Stewart's structure theorem says that for a positive integer $n$, every positive integer $\leq n$ has a ...
1
vote
3answers
25 views

Products of Functions that Don't Depend on Index

Is there a general property of products that allows you to simplify $$ \prod_{i = 1}^n f(x) \, g(i) $$ where $f(x)$ does not depend on $i$? Would it just be $$ f^{n}(x) \prod_{i = 1}^n g(i) $$ ...
1
vote
0answers
31 views

Efficient Cartesian product which ignores classes of elements

Given $n$ sets $X_1,X_2,..,X_n$, and what I am calling an ignore set $I = \{I_1, I_2,..,I_m : \forall i \in I_i, i \in \bigcup X_i\}$. I would like to find the cartesian product $X_1 \times X_2 ...
0
votes
1answer
85 views

Proving the formula for the directional derivatives of the of the sum and dot product of two functions

Define the directional derivative of a function $\textbf{f}$ at $\textbf{c}$ in the direction $\textbf{u}$ by $$\textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) = \lim_{h \rightarrow 0} ...
2
votes
2answers
97 views

Finding $\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$

Recently got this on a test: $$\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$$ Because it's a freshman calculus course, I think we were expected to solve it like a physicist. Taking a look at ...
0
votes
1answer
42 views

Can we express these sets as Cartesian products of two subsets of $\mathbf{R}$?

Let sets $A$ and $B$ be given as follows: $$A := \{ (x,y) \in \mathbf{R}^2 | \ \ x < y \ \ \} $$ and $$B := \{ (x,y) \in \mathbf{R}^2 |\ \ x^2 + y^2 < 1 \ \ \}.$$ Can we express $A$ or $B$ as ...
2
votes
1answer
69 views

$\prod_{k=1}^{n-1}\cos\left(\theta+\frac{k\pi}{n}\right)$

I know that the first one of the following identities holds, yet I don't know the identity of the general case as shown in the title and the bottom of the page. Is there anyone who knows the closed ...
0
votes
1answer
36 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
vote
0answers
31 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
vote
1answer
48 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
votes
0answers
25 views

Product of Consecutive Terms of a Geometric Sequence

Suppose $a_n=aq^n$, where $a>0$ and $q>0$. So $(a_n)_{n=1}^\infty$ is a geometric sequence with positive terms. The product of its consecutive terms, say, $$a_0,a_2,\ldots,a_n$$ equals to ...
0
votes
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 ...
0
votes
0answers
37 views

Product of consecutive integers as alternating sum

In the derivation of the Indian Buffet Process (Indian Buffet Process paper -- see page 1218), we have the following step: $\prod_{k=1}^{K_+}{(K-k+1)} = ...
6
votes
3answers
229 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
votes
2answers
50 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
votes
2answers
236 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: ...
0
votes
1answer
45 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. $$ ...
0
votes
0answers
51 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
votes
1answer
93 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 $F:\Bbb{A}\rightarrow\Bbb{A}$ ...
1
vote
0answers
50 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: ...
0
votes
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
votes
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
vote
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
vote
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
165 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
vote
1answer
218 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
votes
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
votes
1answer
510 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
votes
0answers
58 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
vote
1answer
272 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 ...
0
votes
0answers
42 views

Limit of $\prod_{k=1}^N\left(1-\frac{x^2}{k^2\pi^2}\right)$ [duplicate]

If : $$P(N)=\prod_{k=1}^N\left(1-\frac{x^2}{k^2\pi^2}\right),$$ we get: $$P(N)=\frac{\left(\frac{(x+\pi)}{\pi}\right)_N}{(N!)^2}\left(\frac{1-\frac{x}{N}}{\pi}\right)_N$$ where: $(.)_N$ is the ...
1
vote
1answer
68 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
votes
1answer
66 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 ...
1
vote
1answer
65 views

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

What are canonical injections for co-products in the category Rel?
1
vote
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
130 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 ...
1
vote
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
44 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 ...
1
vote
1answer
48 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 ...