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

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4
votes
1answer
50 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
55 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
votes
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 ...
4
votes
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
votes
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 ...
2
votes
1answer
111 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 $G$ ...
1
vote
1answer
452 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 ...
5
votes
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
votes
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
vote
1answer
299 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
votes
1answer
167 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 ...
15
votes
3answers
4k 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 ...
1
vote
3answers
148 views

Product representations of the factorial function?

Is this the only product representation of the factorial function? $$ {n!} =\prod_{k=1}^{n} k $$
1
vote
7answers
179 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
vote
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 ...
1
vote
1answer
210 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
votes
1answer
488 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
votes
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 ...
0
votes
1answer
26 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
votes
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
votes
2answers
120 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
votes
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
votes
2answers
346 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 ...
1
vote
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 ...
2
votes
1answer
109 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 ...
5
votes
2answers
64 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: ...
10
votes
2answers
274 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
votes
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 ...
0
votes
1answer
64 views

Product rule question about Alphabet

I am trying to understand the product rule and I have a simple example it says, ...
2
votes
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 ...
2
votes
2answers
141 views

Is this a homeomorphism?

Suppose you have a cartesian product of spaces $\prod_{\alpha\in\mathcal{A}}X_{\alpha}$ in the product topology. Choose any $\alpha\in\mathcal{A}$ . Is the following a homeomorphism of a subspace ...
22
votes
4answers
710 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
vote
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
votes
1answer
115 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
votes
1answer
409 views

A product identity involving the gamma function

I have reduced this problem (thanks @Mhenni) to the following (which needs to be proved): ...
1
vote
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
votes
1answer
97 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, \;\;\; ...
2
votes
2answers
798 views

Simplifying a Product of Summations

I have, for a fixed and positive even integer $n$, the following product of summations: $\left ( \sum_{i = n-1}^{n-1}i \right )\cdot \left ( \sum_{i = n-3}^{n-1} i \right )\cdot \left ( \sum_{i = ...
2
votes
1answer
113 views

Identity involving a recursive product

Here is yet another problem related to plane partitions. Given the recursive formula $$ \begin{align*} F(0)&=1,\\ F(r)&=\prod_{i=1}^r\frac{i+2r-1}{2i+r-2}F(r-1). \end{align*} $$ How can we ...
4
votes
4answers
99 views

Big Greeks and commutation

Does a sum or product symbol, $\Sigma$ or $\Pi$, imply an ordering? Clearly if $\mathbf{x}_i$ is a matrix then: $$\prod_{i=0}^{n} \mathbf{x}_i$$ depends on the order of the multiplication. But, ...
3
votes
2answers
76 views

How to define this pattern as $f(n)$

Given a binary table with n bits as follows: $$\begin{array}{cccc|l} 2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ ...
3
votes
1answer
75 views

Show that the following product equals 1 (involves trig)

How can I show that: $$\prod_{k=1}^{n}\left ( 1+2\cos\frac{2\pi .3^{k}}{3^{n}+1} \right )=1$$ Could you please explain to me how to approach this problem? Thank you.
10
votes
3answers
344 views

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?
0
votes
2answers
71 views

General formula for $\prod (x+a_i)$

What could be a general formula for this in terms of $x$ and $a_1,\; a_2\; \ldots\; a_{n-1}, \;a_n$? $$\prod_{i=0}^n(x+a_i)$$ I've tried solving it, but I'm lost at the ...
3
votes
2answers
130 views

Summation and Product Bounds

If I have a sum or product whose upper index is less than its start index, how is this interpreted? For example: $$\sum_{k=2}^0a_k,\qquad \prod_{k=3}^1b_k$$ I want to say that they are equal to the ...
2
votes
0answers
102 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
1
vote
1answer
448 views

Is the product of two measurable subsets of $R^d$ measurable in $R^{2d}$?

Suppose that $E_1,E_2$ are two measurable (Lebesgue) subsets of $R^d$. Define $E=E_1\times E_2=\left\{(x,y)|x\in E_1, y\in E_2\right\}$. Can we say that $E$ is a Lebesgue measurable subset of ...
2
votes
1answer
61 views

Proving a poset is atomic

A poset $(X,\le) $ is atomic if it has both a smallest and largest element, it is graded ,and every element $x$ of $X$ is the join $x_1\vee \dots\vee x_n$ of some elements of $X$ (also written as ...
7
votes
2answers
219 views

Showing an indentity with a cyclic sum

Let $n\geqslant2$, and $k\in \mathbb{N}$ Let $z_1,z_2,..,z_n$ be distinct complex numbers Prove that $$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j ...
6
votes
1answer
149 views

Is there another way to write the product $\prod_{k=0}^n\left(k+\alpha\left(-1\right)^{k+1}\right)$?

I have the following expression $$ \prod_{k=0}^n \left(k + \alpha(-1)^{k+1}\right), $$ which is, for example, $(0-\alpha)(1+\alpha)(2-\alpha)$ for $n = 2$. Is there a way to write this using ...