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Questions tagged [products]

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

0
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2answers
16 views

Cartesian product with three sets

I was given a question that asks "Let A={1,2} ,B={x,3,y},C={2,y}. Find A x B x C and C x B x C " I have an example that I could go by but i'm not sure what they were multiplying together to get it. If ...
0
votes
1answer
53 views

Is this sum constant for n?

Hi I can prove that this sum is constant in $n\in \mathbb{N}$. However my proof is very long (a few pages with probability involved). Does anyone see a simple proof. The sum in question is (a q-series ...
2
votes
3answers
46 views

Check proof that $\prod_{k=1}^n(1+{1\over a_k})$ is bounded if $a_{n+1} = (n+1)(a_n + 1)$ and $a_1 = 1$, $n\in \mathbb N$

Let $n \in \mathbb N$ and: $$ \begin{cases} a_1 = 1 \\ a_{n+1} = (n+1)(a_n + 1) \end{cases} $$ Prove that $$ x_n = \prod_{k=1}^n\left(1+{1\over a_k}\right) $$ is a bounded sequence. Obviously ...
0
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0answers
11 views

If $f(t)=\Pi_{i=1}^{d-1}\left(1+tb_{i}\right)^{\alpha}$ is concave on $0<t<1$, then $\alpha (d-1) \leq 1$

Let's $$f(t)=\prod_{i=1}^{d-1}\left(1+tb_{i}\right)^{\alpha}$$ with $b_{i} > 0$. I want to show that if $f(t)$ is concave on $0<t<1$, then $\alpha (d-1) \leq 1$ I tried since $b_{i}&...
-1
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1answer
18 views

Cartesian product of 2 dimensional

Let $R=\{(1,1),(2,2),(3,2),(4,1)\}$. Then how can I calculate $R\times R$?
1
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1answer
31 views

Is $NK=KN$ still even if only one of them is normal but both are still subgroups?

In this question Prove that the product $NK$ of two normal subgroups $N$ and $K$ of a group $G$ is a normal subgroup of $G$, and $NK=KN$., it is proved that $NK=KN$ if both $N$ and $K$ are normal ...
1
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1answer
75 views

How to calculate the following product?

How can we compute $$\prod_{i=n}^{1}\cos^{2}\left(\frac{2^{i}}{2^{n+1}}x\right)$$ for $0<x<\pi$ ? Attempt: $$ \begin{align*} \prod_{i=n}^{1}\cos^{2}\left(\frac{2^{i}}{2^{n+1}}x\right) & ...
-2
votes
1answer
29 views

How to show that $\prod_{i=1}^n a_i = \prod_{i=1}^{r} a_i \times \prod_{i=r+1}^{n} a_i$?

I need to prove this multiplication property. Is it correct? Suppose that $1\le r\le n$. Show that: $$\prod_{i=1}^n a_i = \prod_{i=1}^{r} a_i \times \prod_{i=r+1}^{n} a_i. $$
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0answers
27 views

Extreme-mega-ultra-crazy hypergeometric functions

I've been lusting over hypergeometric functions, and came up with some questions. Here goes. I've defined the following functions, and I want to know if there are any closed forms for them. $$f_1(2;...
0
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0answers
25 views

Closed form formula for an expression involving sums of products.

I am stuck trying to simplify or find a closed form formula for the following expression: $$\sum_{i=0}^{n-1} (\prod_{y=i+1}^{n-1}\frac{y+1}{y+2} \div \prod_{x=i+1}^{k-2}\frac{x+1}{x+2})$$ The ...
1
vote
0answers
33 views

Closed form for $_2 F_2(\alpha,3\alpha;2\alpha,4\alpha;x)$

I came up with a potentially interesting hypergeometric function. I know that $$_2 F_2(\alpha,3\alpha;2\alpha,4\alpha;z)=\sum_{n=0}^{\infty}\frac{(\alpha)_n(3\alpha)_n}{(2\alpha)_n(4\alpha)_n}\frac{z^...
1
vote
1answer
22 views

Product Notation Conventions

If a product has the following notation, does that mean the product has zero terms? If there is an expression after this symbol, is the entire quantity, with the product symbol included, equal to zero?...
3
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4answers
35 views

Equivalent Capital Pi Notation Expressions

So I was working on a probability question and then this expression came up. When I consulted the answers, I struggled to understand exactly how I would get from one expression to the other myself. ...
1
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1answer
45 views

$\prod_{i=1}^{n}(x+\alpha_i)$ series expansion

Let $I=\{1,2,...,n\}$ for any $n\in \Bbb N$. Assume that $\forall i\in I, \alpha_i\in\Bbb R$. What is the series expansion for $$p_n(x)=\prod_{i\in I}(x+\alpha_i)$$ I've noticed that $p_n(x)=xp_{n-1}(...
0
votes
2answers
23 views

Cardinality of product of subsets of a group

Let $A, B$ be finite subsets of a group $G$ (not necessarily finite). Is it true that $|AB| = |BA|$? More generally, is it true that $|ABC| = |ABC| =\cdots$ any permutation of three elements, if $C$ ...
1
vote
1answer
21 views

Is there a way to distribute a derivative into a product of arbitrary bounds like a summation?

What I am interested in is $$ \frac{d}{dx}\prod_{a=1}^{b}f_{a}(x). $$ I know that a derivative can easily be distributed into a summation, but what about an arbitrary product?
0
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0answers
14 views

Product of groups and categories

If G is a group, then {G} is a group (class) with an object whose morphemes are elements of G, prove that if G is not obvious(evident)(|G|>or =2), then the product of {G, G} is not present(exist). (...
0
votes
0answers
11 views

prove that relative interior preserves product

Let $C_1\in R^n$ and $C_2\in R^m$ be convex sets. I want to prove that $rint(C_1\times C_2)=rint(C_1)\times rint(C_2)$. I know that this question is already asked and answered, but I want to prove it ...
0
votes
0answers
50 views

Closed form of this product or approximate?

What is the closed form of this product: $$\prod_{i=1}^{k-1}\left(1-e^{-a(b- ic)^2}\right)$$ where $a,b,c$ are constants?
0
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0answers
19 views

Proper class of operations

Usually, a collection of operations forms a set. But I've heard of an example where it is (must be) a proper class. Namely, that $\mathbb{CompHauss}$ becomes a variety if proper class of operations ...
7
votes
1answer
153 views

For positve $a$, $b$, $c$, $d$ with $a+b+c+d\leq 1$, prove that $\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq255\cdot a b c d .$

Let $a,b,c,d\in\mathbb R_+$ such that $a+b+c+d\leqslant1$. Prove that$$ \sqrt[4]{\smash[b]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}}\geqslant255·abcd. $$ My observations: I can see that all of $a,b,c,d$ are ...
3
votes
3answers
331 views

How do I calculate $(A\times B)^2$ for $A=\{1,2\}$ and $B=\{a,b,c\}$?

Let $A = \{1,2\}$ and $B = \{a,b,c\}$, find $(A \times B)^2$. I found $(A \times B) = \{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\}$ But how do I find $$\{(1,a),(1,b),(1,c),(2,a),(2,b),(2,c)\} \...
1
vote
0answers
38 views

Categorical product cancellable

It is pretty easy to prove that $A\cong B\Rightarrow A\times X\cong B\times X$ (assuming the products exist). However, I haven't been able to prove the other direction, that is, $A\times X\cong B\...
5
votes
2answers
110 views

An equality between a product and a combinatorial sum

I'm trying to prove the following identity (of which I numerically verified the truth) : $$\text{For every $n\in\mathbb{N}^*$ and $\alpha \in \mathbb{R}\setminus\lbrace-2k\text{ }|\text{ }k\in\mathbb{...
0
votes
1answer
44 views

Infinite sum of $\sum_{n=1}^\infty \prod_{k=1}^n \frac{2k+1}{4k}$ [closed]

Here I have this sum, $\displaystyle \sum_{n=1}^\infty \displaystyle \prod_{k=1}^n \dfrac{2k+1}{4k}$. I have no idea how to sum this up. Any help would be appreciated!
0
votes
2answers
42 views

Product of Jacobians and chain Rule

I've got a question on how to achieve this proof correctly (2nd year vector calculus) : Assuming that u = u(r, s) , v = v(r, s) , r = r(x, y) and s = s(x, y) are $C^1$ functions, prove that $$\frac{...
2
votes
3answers
78 views

About the limit of $\prod\limits_{k=2}^n (1-\frac{1}{k^3})^k (1+\frac{1}{k^3})^{1-k}$

Question: Does a formula exist for $\enspace\displaystyle \prod\limits_{k=2}^n (1-\frac{1}{k^3})^k (1+\frac{1}{k^3})^{1-k}\enspace$ so that it can be seen that the limit is $\,\displaystyle\frac{...
3
votes
1answer
40 views

Can $1+\prod_{i=1}^n a_i$ be prime for all $n$?

This is inspired by Show, by computing several values, that there are composite numbers in this sequence. Is there an increasing sequence of positive integers $(a_i)|_{i=1}^{\infty} $ such that $1+\...
0
votes
1answer
22 views

Finding products in groups of three

If a triple is defined as (a,b,c) such that $a \times b \times c = n$ What is the best way to find these triples for a given $n$ (1,2,3) is the same triple as (3,2,1) as is (2,1,3) etc.. I need to ...
0
votes
0answers
16 views

How to find the Jacobian of a matrix vector product?

If $H(t)$ is a given two dimensional Hamiltonian matrix , $Y$ is a vector (spinor). Defined is the function $F(t,Y)= H(t)Y$. What is the Jacobian of F(t,Y). In some numerical method I need to find ...
1
vote
2answers
103 views

How to prove this inequality using AM-GM?

Suppose $a,b,c$ are positive real numbers. Then prove that $$\Big(\frac{a+b}{2}\Big)\Big(\frac{b+c}{2}\Big)\Big(\frac{c+a}{2}\Big)\ge\Big(\frac{a+b+c}{3}\Big)\Big(abc\Big)^\frac{2}{3}\tag{*}$$ My ...
0
votes
1answer
23 views

Inner product of multivariable functions

How is $$\langle f, g\rangle \text{where} f(x,y), g(x,y)$$ defined, where $\langle \cdot ,\cdot \rangle$ means the inner product of $f$ and $g$? (I primarily searched How can we define the Inner ...
0
votes
0answers
16 views

Interpret the 'x' symbol to be multiplication or cross product when referencing 2 complex numbers?

For two complex #s, A = 2 + j*16 & B = 7 + j*2, I understand how to multiply them: Rectangular - (2 + j*16)*(7 + j*2) = 14 + 4*j + 112*j + 32*j^2 = -18 + 116*j Polar - (2*sqrt(65)∠82.875°)*(sqrt(...
1
vote
1answer
32 views

What is the notation for a partial factorial?

All of this happened so long ago for me, things are a little rusty. I’m calling this a partial factorial because I don’t know what else to call it. What is the notation for a product of the form: <...
5
votes
3answers
237 views

If $A=1$, $B=2$, etc, then what word, treated as a product of its letters, has value closest to $1000000$?

Suppose that a product $n$ is the product of the numbers corresponding to its letter, eg. $A = 1$, $B = 2$, etc. What is the word that has a product close $1000000$? Here's some examples: $$\begin{...
1
vote
1answer
32 views

About the value of $\prod\limits_{\ell=0}^j((k+1)^2-(\ell+1)^2)$ where one omits the factor $\ell=k$

I have a question regarding this answer. At one point, Alex Francisco writes at the end that $$\prod\limits_{\substack{0 \le \ell \le j \\ \ell \neq k}} ((k + 1)^2 - (\ell + 1)^2) = \dfrac{(-1)^{j - k}...
3
votes
2answers
61 views

Turning a product into a sum

Is it possible to change $$\prod_{i=1}^n(1+2a_ib_i),$$ where all elements are contained in an unital associative algebra generated by $a_i,b_i$, $i=1,...,n$, such that $a_ib_i=-b_ia_i$, into a sum ...
1
vote
0answers
28 views

$\mathcal{L}(V\times\cdots\times V_m,W)$ and $\mathcal{L}(V_1,W)\times\cdots\times\mathcal{L}(V_m,W)$ are isomorphic.

Is the following argument? Suppose $V_1,\dots,V_m$ are vector spaces. Prove that $\mathcal{L}(V\times\cdots\times V_m,W)$ and $\mathcal{L}(V_1,W)\times\cdots\times\mathcal{L}(V_m,W)$ are ...
2
votes
1answer
63 views

A product involving complex numbers

Given $ω^3=1$ but $ω≠1$ how would you prove that, $\displaystyle \prod_{m=1}^{2n} (1-ω^{2^m}+ω^{2^{m+1}})=2^{2n}$ ? I tried but made no progress tackling this.
1
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0answers
44 views

Relative prime density of $f(n)$

Relative prime density of $f(n)$ Definition Number of primes of $f(n)$ Using the prime counting function $\pi(n)$, we can define the number of primes $\pi_{nbr}$ of a function $f(n)$ from $n=1$ to $...
1
vote
1answer
19 views

Sum of partition's product modulo 5 up to 41

If we define $S(n)$ as $4=3+1=2+2=2+1+1=1+1+1+1$ $S(4)=3\cdot1+2\cdot2+2\cdot1\cdot1+1\cdot1\cdot1\cdot1=3+4+2+1=10$ $5=4+1=3+2=2+2+1=2+1+1+1=1+1+1+1+1$ $S(5)=4\cdot1+3\cdot2+2\cdot2\cdot1+2\cdot1\...
4
votes
4answers
71 views

Evaluating a product $\prod_{n=0}^{100}\left(1+\dfrac{1}{a^{2^n}}\right)$

How would you evaluate this product? $$\displaystyle \prod_{n=0}^{100}\left(1+\dfrac{1}{a^{2^n}}\right)$$ I know one way in which multiplying the product by $\left(1-\dfrac{1}{a}\right)$ does the job ...
2
votes
1answer
104 views

Max product for a list of digits

There is a list of digits, lets say: "1", "1", "9", "9" And a multiply operator; how to make the max product. The answer for above example is: ...
5
votes
2answers
44 views

On Abelian group product of a Free and a Finite group

I'd need help with this problem: Let $H$ be an abelian group. Let $T \leq H$ and $T' \leq H$ be finite subgroups of H. Let $F \leq H$ and $F' \leq H$ be free subgroups of H. Suppose $H = T \times F =...
0
votes
1answer
36 views

Can you turn a Taylor series into a product and is there any application for it?

In a lot of older texts I see a lot of constants and functions represented by infinite products like $$ \frac{a}{a-z}= \prod_{n=1}^{ \infty}e^{ \frac{1}{n}( \frac{c}{z})^{n}},$$ but nowadays all I ...
6
votes
4answers
107 views

Expanding product of binomials $(z^k + z^{-k})$

Suppose $z$ is a complex number, and consider the product $$f_m(z)=\prod_{k=1}^m \left(z^k + \frac 1 {z^k} \right),$$ for $m = 1,2,\dots$ . Of course, one should be able to expand this into a sum of ...
0
votes
1answer
76 views

What's equal this:$\int_{0}^{\infty}\prod_{k=0}^{n}{\cot^{-1}(x^k+\frac{1}{x^k})}dx$?

Evaluation for some integer $k$ using wolfram alpha show that this integral $\int_{0}^{n}\prod_{k=0}^{n}{\cot^{-1}(x^k+\frac{1}{x^k})}dx$ converges fast and deacreasing to $0$ , Now my question ...
-1
votes
1answer
33 views

How many Scalers can be built using three different unit vectors? [closed]

I have three unit vectors in a problem: $\hat{t}= (\cos(t),0,\sin(t)),$ $ \hat{m}= (0,0,1),$ $\hat{n}= (\sin(th),0,-\cos(th)).$ I know the solution for the problem is: $(-\sin(2t)+ 5 \sin(2t-4th)+...
1
vote
3answers
48 views

Strengthening an inequality of exponential series

A question posits that, show: $$(1+x_1)(1+x_2)\ldots(1+x_n)\\ \leq 1+\dfrac{S}{1!}+\dfrac{S^2}{2!}+\ldots\dfrac{S^n}{n!}$$ Where $S=\sum x_i$, $x_i\in\mathbb{R}$ Now, the RHS looked like the ...
0
votes
1answer
26 views

Product of matrices multiplying each row by corresponding column

I was wondering if there exists a certain type of matrix multiplication that just multiplies row $i$ by column $i$. For example let be $$ A=\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ ...