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

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1answer
30 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 ...
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1answer
254 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
305 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
83 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 ...
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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
619 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
65 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
394 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
80 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 ...
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1answer
42 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
70 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
79 views

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

What are canonical injections for co-products in the category Rel?
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1answer
32 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
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1answer
133 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
30 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
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1answer
51 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
59 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 ...
0
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0answers
79 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 ...
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4answers
98 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
121 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 ...
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1answer
113 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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2answers
2k views

Proving the AM:GM inequality [duplicate]

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
28 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
312 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
169 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 ...
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3answers
5k 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
148 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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7answers
180 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
110 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
231 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 ...
11
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1answer
526 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
61 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 ...
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1answer
28 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
75 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
122 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?
3
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2answers
109 views

Why is $V_{4}$ the semi direct product of $Z_{2}$× $Z_{2}$

I'm trying to understand what is a semi direct product , so by the definition semi-direct product of G , I'd need two groups , $N$ and $H$ , where : $H∩N$ = {e} $H \cdot N$ = $G$ If $H=N=Z_{2}$ ...
4
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2answers
348 views

General expression for $\sin(2^n x)$

Are there general expressions for $\sin(2^n x)$ and $\cos(2^n x)$ that only involve $\sin x$ and $\cos x$, and that moreover involve only polynomial (in $n$) number of terms? Edit: $2^n$ is not ...
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0answers
34 views

Efficient way to compute $n$ products of $n$ numbers

Say I have a set of $n$ numbers ${a_1, ..., a_n}$. I want to compute $n$ products, where the $i$th product is defined as the product of all elements in the set, except $a_i$. For example, for $n=5$, I ...
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1answer
111 views

approximation to “generalized binomial coefficient”

What is the limit, when $n$ goes to $\infty$, of the following product, when $0 \leq a \leq 1$? $$ {{1-a} \over 1}\cdot {{2-a} \over 2} \cdot {{3-a} \over 3} \cdot\ldots\cdot {{n-a} \over n} $$ When ...
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2answers
66 views

Evaluating $\prod_{r=1}^{n} (2r+1)$

Could someone please help me as to how I'd go about evaluating: $$\prod_{r=1}^{n} (2r+1)$$ I have that written out, it is: $$1 \cdot 3 \cdot 5 \cdots (2n-1) \cdot (2n+1)$$ furthermore: ...
11
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2answers
276 views

Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

The current issue (vol. 120, no. 6) of the American Mathematical Monthly has a proof by probabilistic means that $$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k} $$ for ...
3
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1answer
168 views

Partition Proof

Let $\lambda$ be a partition of $N$ of rank $r$. How can I show that: $$\sum_wx^\lambda(w)=f^\lambda(-1)^{t(\lambda)}\prod^r_{i=1}(\lambda_i-1)!(\lambda'_i-1)!$$ where $w$ ranges over all ...
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1answer
65 views

Product rule question about Alphabet

I am trying to understand the product rule and I have a simple example it says, ...
2
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1answer
83 views

How to calculate a bound for this product?

Consider the following product: $$ \prod_{i=1..n} {\left(1 - {1 \over 2^i}\right)} $$ A numeric calculation, up to $n=20$, gives $0.288788370496567$. But how can I calculate its limit when $n$ goes ...
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4answers
720 views

Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
1
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1answer
7k views

Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2} $ aka the square root of the dot product... ...
0
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1answer
118 views

Expression for sum of $k$-products of $n$ variables

Given $n$ variables there are $n \choose k$ different terms that are the product of $k$ different variables. For example, in the case that $n = 3$, the $k$-products of the variables $x_1, x_2, x_3$, ...
3
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1answer
416 views

A product identity involving the gamma function

I have reduced this problem (thanks @Mhenni) to the following (which needs to be proved): ...
1
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1answer
94 views

faithful irreducible representation of $A_{4} \times Q_{8}$

Construct a faithful irreducible representation of the group $A_4 \times Q_8$ $A_{4}$ is the alternating group $Q_{8}$ is the quaternions
4
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1answer
98 views

Distribution of Digit Products

A digit product $P(n)$ of a natural number $n$ is given by the product of its decimal digits. For example: $$P(1234) = 24,\;\;\; P(24) = 8,\;\;\; P(8) = 8$$ $$1\times2\times3\times4 = 24, \;\;\; ...