Questions tagged [products]
For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.
1,821
questions
-2
votes
0
answers
51
views
Can I write $\prod_{i=1}^{n}(y_i+\lambda z_i) = \prod_{i=1}^{n}y_i + \lambda \prod_{i=1}^{n}z_i$? [closed]
Is product notation distributive. As in, can I write the following? $$\prod_{i=1}^{n}(y_i+\lambda z_i) = \prod_{i=1}^{n}y_i + \lambda \prod_{i=1}^{n}z_i$$
- 959
0
votes
0
answers
16
views
Bound of a ratio of summations and products of terms of bounded ratios
Given $D_1, D_2 \in \mathcal{S}$ and given a space $\mathcal{M}$ such that $\mathcal{S} \subset \mathcal{M}$, I have the following function:
$$
\frac{\sum_{\mathcal{S}_{i} \in \mathcal{M}} \left[ f(...
1
vote
1
answer
135
views
+50
Prove or (dis)prove that : $f(x,y)\geq 0$
I ask for a proof of :
Let $x,y\in(-0.9,0.9)$ then it seems we have :
I have conjecture for and $x,y\in(-1,1)$ :
$$f(x,y)=\ln\left(\frac{(1+xy)^{2}(1+x^{2})}{(1-xy)^{2}(1-x^{2})}\right)-\left(x\...
- 3,243
1
vote
1
answer
52
views
If $N$ and $K$ are normal subgroups of a group $G$ such that $G=MN$ and $M\cap N=\langle e \rangle$ then $G=M\times N$.
The following is an exercise in Hungerford's abstract algebra text.
If $N$ and $K$ are normal subgroups of a group $G$ such that $G=MN$ and $M\cap N=\langle e \rangle$ then $G=M\times N$.
If $G=S_3$ ...
- 2,195
0
votes
0
answers
13
views
Formula for $\prod_{i=2}^{n-1}\left( \prod_{j=1}^{m} \left( \left((n-i)^2 + 1\right) \times m - j \right) \right)$?
I am computing the complexity of an algorithm with inputs $n$ and $m$ and I find the following result (number of iterations): $\prod_{i=2}^{n-1}\left( \prod_{j=1}^{m} \left( \left((n-i)^2 + 1\right) \...
- 21
1
vote
1
answer
50
views
Convergence of an infinite product using logarithm
Suppose we have $(a_n)_{n\geq 1}$ is a positive real sequence, and we know that $\sum\log a_n$ converges. How can we conclude that the product
$$
\prod_{n\geq 1}\sqrt{\frac{2}{a_n+a_n^{-1}}}
$$
...
- 153
0
votes
1
answer
23
views
Simplifying the product of $(x-x_j)$'s when $x_k$ has multiplicty $\geq 2$
Let $w(x)$ be defined as such: $w(x) = \prod_{j=0}^{n}(x-x_j)$, where $x_j$'s are distinct real numbers, $j=0,1,...,n$.
Suppose there exists exactly one point $x_k$, $k \in {0,1,...,n}$ such that $x_k$...
- 814
2
votes
2
answers
84
views
Equation involving double sum and product
I am a physicist who needs your help. I am currently working on a problem in relativity and to verify a result that I derived I should prove this terrible looking equation:
$$\binom{l}{2\alpha}\prod_{...
- 439
1
vote
1
answer
74
views
curious GoldenRatio identity [closed]
I would like to verify the following identity but I don't know how mathematics says that it is equal to numerically.
$$\prod _{k=0}^{\infty } \sqrt{\frac{\phi ^{2^{-k-1}} \left(\phi ^{2^{-k}}+1\right)}...
- 19
0
votes
0
answers
31
views
How to calculate this limit by stolz theorem
Let $f(x)$ be a positive function such that $\displaystyle\lim_{n\to \infty} \frac{f(n)}{n}=a>0.$
The question is how to calculate the following limit:
$$\lim_{n\to \infty}\sqrt[n+1]{\prod_{k=1}^{n+...
- 718
2
votes
2
answers
87
views
How to prove $\prod_{k=1}^n (1+a_k)\leq \sum_{j=0}^n\left(\sum_{k=1}^n a_k\right)^j$
If $a_{i} \geq 0$ , then $(1+a_{1})(1+a_{2})\cdots (1+a_{n}) \leq 1+(a_{1}+a_{2}+\cdots+a_{n})+(a_{1}+a_{2}+\cdots+a_{n})^{2}+\cdots+(a_{1}+a_{2}+\cdots+a_{n})^{n}$
I found it in the book Theory and ...
- 781
1
vote
1
answer
77
views
closed form of $\prod_1^n (x+k)$
Is there a closed form for the product
$$
\prod_{k=1}^n (x+k)
$$
I'm trying to find a nice formula for
$$
\Gamma(z+n) = f(n)\Gamma(z)
$$
for some appropriate f.
- 13
0
votes
1
answer
43
views
Difference in two products of prime factorizations
Let $\Phi(n)=\{p_1, p_2, ..., p_k\}$ be the set of prime factors of a number $n$. How does
$$
p_1(n) = \prod_{p_i\in\Phi(n) \\ 1 \le i \le k}{p_i}
$$
compare to
$$
p_2(n) = \prod_{p_i\in\Phi(n) \\ 1 \...
- 100
3
votes
1
answer
92
views
Proving the uniqueness of a map
Here is the question I am trying to solve:
(Tensor product of coalgebras) Let $(C, \Delta, \varepsilon)$ and $(C', \Delta ', \varepsilon ')$ be coalgebras. Show that the linear maps $\pi: C \otimes C' ...
- 1,299
0
votes
1
answer
37
views
How to prove $\prod_{k=0}^n\left(2-\frac{2k+1}{n}\right)=-\frac{(2n)!}{2^n n^{n+1}n!}$.
To finish a proof, I am stuck on the steps of getting from $$\prod_{k=0}^n \left(2-\frac{2k+1}{n}\right)$$ to the form $$-\frac{(2n)!}{2^n n^{n+1} n!}.$$
If it helps, the entire question as follows: ...
- 3
5
votes
2
answers
136
views
Prove that $\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$
For arbitrary $x$ and $1\leqslant m\leqslant n$, prove the following:
$$\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$$
I'm looking for a proof that ...
- 546
1
vote
1
answer
40
views
Find $f(\frac{2\pi k}{2^n \pm1})$ given $f(x)=\prod_{i=1}^{n-1}[2\cos(2^{i-1}x)-1],n\geq1$.
We are required to find $f(\frac{2\pi k}{2^n \pm1})$ given
$f(x)=\prod_{i=1}^{n-1}[2\cos(2^{i-1}x)-1],n\geq1$
Let's start by removing the product notation
$\Rightarrow f(x)=(2\cos x-1)(2\cos 2x-1)(2\...
0
votes
0
answers
37
views
Calculate Likelihood Function for Gamma/Weibull Distribution
I'm trying to calculate the likelihood function (not log-likelihood) for a random sample of n observations from a Gamma Distribution and a Weibull Distribution and I'm struggling to work with the pi ...
- 320
1
vote
0
answers
60
views
Is there a nonzero commutative and associative product preserving the symmetric positive semidefinite matrices?
Question 1. For $d > 1$, is there a nonzero commutative and associative product on the real $d \times d$ symmetric matrces $S^d$ preserving the real symmetric $d \times d$ positive semidefinite ...
- 4,446
1
vote
1
answer
29
views
Hasse Diagram for Cartesian Product of Three Sets
I have to draw Hasse diagram for three sets in the picture.
Is my drawing correct?
- 148
2
votes
2
answers
119
views
Finding the derivative of $y = x^{(x+1)(x+2)(x+3)(x+4)\ldots(x+n)}$. [closed]
I'm trying to find the derivative of this function with respect to $x$:
$$y = x^{(x+1)(x+2)(x+3)(x+4)\ldots(x+n)}$$
I was thinking about using $\ln$ to solve this, but I'm not sure if that's the right ...
- 55
2
votes
1
answer
167
views
Which number is greater A or B?
Let :
$$I_k=\int_{0}^{1}\left(\prod_{n=1}^{k}\left(1+\arctan\left(\left(\frac{y}{4n^{2}}\right)\right)\right)\right)dy$$
And :
$$h\left(x\right)=\int_{0}^{1}\left(\prod_{n=1}^{\operatorname{floor}\...
- 3,243
0
votes
0
answers
29
views
Is there some terminology for the "opposite" of a subdirect subgroup
We say $H \leq G^{n}$ is a subdirect subgroup (in the context of groups) of $G^{n}$ if each projection map (restricted to $H$) is surjective.
Is there a terminology for $H \leq G^{n}$ s.t. NONE of the ...
- 96
1
vote
3
answers
177
views
$\lim\limits_{x\to \infty}[f(x)-f(x-1)]\overset{?}{=}e$
Let :
$$f\left(x\right)=\int_{0}^{\lfloor x\rfloor}\prod_{n=1}^{\lfloor x\rfloor}\frac{\left(y+2n\right)\ln\left(y+2n-1\right)}{\left(y+2n-1\right)\ln\left(y+2n\right)}dy$$
Conjecture:
$$\lim_{x\to \...
- 3,243
0
votes
1
answer
77
views
Clever ways to expand $\prod_{i=1}^{n}\left(1+\frac{n^2}{i^2}\right)\left(1+\frac{i^2}{n^2}\right)^\frac{n^2}{i^2}$?
This is a continuation from this thread.
From my work in the link above I found the following.
$$\int_{0}^{\infty}\ln\left(1+\frac{1}{x^2}\right)dx=\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^n\ln\left(\...
- 185
8
votes
0
answers
199
views
Help me to get deeper understanding of Euler's proof of his Arithmetical Theorem
With distinct numbers $a_1, a_2, \ldots, a_n$, let's denote the products of the differences of each of these numbers with the each of the rest of them by the following principle:
\begin{align}
(...
- 546
1
vote
1
answer
47
views
Product of $\prod_{n=0}^{k-1} \frac{-1}{(4n+4)(4n+3)}$
Is there any way of expressing $$\prod_{n=0}^{k-1} \frac{-1}{(4n+4)(4n+3)}$$ as some special function? I can get factor some things out and get $$\frac{(-1)^k}{4^kk!}\prod_{n=0}^{k-1} \frac{1}{4n+3}$$ ...
- 50
1
vote
2
answers
50
views
What computational shortcut finds the sum all possible products given any list of n random real numbers taken r at a time? Here's what I tried...
I have the following computational shortcuts for any list of $n=4$ quantities taken r at a time. My goal is to do this for lists of any length. Taken r at a time, what function can similarly output ...
- 45
0
votes
1
answer
47
views
Is it possible to evaluate $\prod_{j=1}^{m}\sin(\sqrt{j}x)$
Basically wondering if it's possible to evaluate or approximate products like,
$$\prod_{j=1}^{m}\sin(\sqrt{j}x)$$
I thought that
$$\sin^{m}(\sqrt{0.5m}x)$$
may be a reasonal approximation, since it ...
- 233
7
votes
1
answer
321
views
How to prove the following discovery of Euler?
There exists a series of formulas.
\begin{align*}
\ & \dfrac{1}{(a-b)(a-c)}+\dfrac{1}{(b-a)(b-c)}+\dfrac{1}{(c-a)(c-b)} = 0, \\
\ & \dfrac{a}{(a-b)(a-c)}+\dfrac{b}{(b-a)(b-c)}+\dfrac{c}...
- 546
1
vote
1
answer
37
views
Coefficient of $x^k$ in the product of multiple polynomials
It is well know that if we have two polynomials $f$ and $g$ such that $$f(x)=\sum_{i=0}^nf_ix^i,\quad g(x)=\sum_{j=0}^mg_jx^j$$
then the coefficient of $x^s$ in the product $f(x)\cdot g(x)$ is exactly ...
- 3,775
3
votes
1
answer
90
views
Intersection of projective varieties
Is there any way to study varieties of the form below
$$
X=V\left(\{ f_i,g_j|i\in I,j \in J \right\}) \subset \mathbb{P}^{m+n+1},
$$
where
$$
f_i \in k\left[ x_0, \cdots , x_m \right],\ g_j\in k\left[...
- 33
3
votes
1
answer
44
views
$G \times H$ regular $\implies G $ and $H$ regular?
I know that if two graphs $G,H$ are regular then their Cartesian product is also regular. But I never heard of the veracity of the converse. Is is true ? I think yes by the following argument but I am ...
- 1,471
0
votes
0
answers
22
views
A fininite sum involving products of the $q$-shifted factorial type
During Christmas I played with partial sums of convergent geometric series and I was able to deduce the identity
$$
\sum_{i=0}^K\frac1{q^{i(s+1)}\cdot\prod_{j=1}^{K-i}(q^j-1)\cdot\prod_{j=1}^i(q^{-j}-...
- 503
0
votes
1
answer
59
views
Recursive formula for the derivative of the product
Let $f(x)$ be a function such that $f'(x) = f(x)g(x)$. Is there a general way to express the $n^{th}$ derivative of $f(x)$ such that
$$
f^{(n)}(x) = f(x)h(x),
$$
where $h(x)$ is a function of the ...
- 11
2
votes
4
answers
297
views
How to find the constant $C$ such that $f(x)\geq Cx$
Problem :
Define for strictly positive $x$ :
$$f\left(x\right)=\left(\prod_{k=1}^{\operatorname{floor}\left(x\right)}\left(1+\sum_{n=1}^{k}\frac{1}{k\cdot2^{n}}\right)\right)$$
Does there exists a ...
- 3,243
15
votes
0
answers
291
views
Is this just a coincidence? Expectation of product of areas in unit circle equals $\pi^2/6$, the answer to the Basel problem.
Draw line segments from the centre of a unit circle to two uniformly random points on the circle, forming two regions of area $A_1$ and $A_2$.
It is easy to show that the expectation of the product ...
- 8,500
4
votes
1
answer
90
views
Proving $\prod_{j=1}^n(1-\prod_{i=1}^m\sin^2x_{ij})+\prod_{i=1}^m(1-\prod_{j=1}^n\cos^2x_{ij})\geq1$, for real numbers $x_{ij}$
I have been struggling to solve the following problem which seems to be some kind of generalized trigonometric Pythagorean identity:
Let $x_{ij}$ $(1 \leq i \leq m$, $1\leq j \leq n)$ be real numbers,...
- 51
-2
votes
1
answer
67
views
Proving $\prod_{k=1}^{N} \left(1 - \frac{1}{k+1}\right) = \frac{1}{1+N}$ [closed]
It's been a long time since I've done this sort of thing, so can't remember how to solve this or the specific key terms to look it up and check for duplicate answers.
$$\prod_{k=1}^{N} \left(1 - \frac{...
- 680
-2
votes
1
answer
56
views
Prove and generalize $\cos\frac\pi9\cos\frac{2\pi}9\cos\frac{3\pi}9\cos\frac{4\pi}9=\frac1{16}$ [duplicate]
Is there a simple proof for the following?
$$16\cos\frac\pi9\cos\frac{2\pi}9\cos\frac{3\pi}9\cos\frac{4\pi}9=1$$
Is this statement valid for any odd number (not only 9)?
Is $\cos x=\frac{e^{ix}+e^{-ix}...
0
votes
0
answers
53
views
Difficulty in solving an exercise about why product/coproduct is needed for 'words on the alphabet'.
The following question is taken from Arrows, Structures and Functors by Arbib and Manes.
For each set $A$, the set $A*$ of all 'words on the alphabet $A$' may be defined as
$$A*=\coprod_{n=0}^{\...
- 2,195
0
votes
0
answers
33
views
Looking for a closed form expression of $\prod_{n=1}^{m}(n!)^n$ or $\prod_{n=1}^{m}n^{-n^2}$
I'm working on some math in my spare time and wanted to see if it were possible to find $\prod_{n=1}^{m}(n!)^n$ as a closed form expression.
I was able to work through it to get it down to $$\prod_{n=...
2
votes
0
answers
79
views
The result of Big Pi notation "without any element". [duplicate]
This is such a simple question but I couldn't find the answer on the internet. What is the default result of the Big Pi notation when it happens to be applied to an empty set? Is it 1 or 0, or even ...
- 99
1
vote
1
answer
30
views
Closed-form value of a product $\prod_{i = 2}^n (1 + \frac{p}{i})$ with $0 \leq p \leq 2$
Let $f(n+1;p) = \prod_{i = 2}^n (1 + \frac{p}{i})$, where $0 \leq p \leq 2$ and $n \geq 2$ with $f(2;p)=1,\forall p$.
We have $f(n;0) = 1$, $f(n; 1)=n/2$.
I also can see that $f(n; 2) \leq n^2/4$.
Is ...
- 1,154
3
votes
1
answer
122
views
How can I prove that this matrix is idempotent?
I have the following matrix
$$A=\begin{equation}
\begin{pmatrix}
0 & a & -b\\
-a & 0 & c\\
b & -c & 0
\end{pmatrix}
\end{equation}$$
I have to prove that $M=A^2+I$ is ...
- 163
0
votes
0
answers
39
views
Geometric Mean To Calculate Event Probability vs Product Of Outcomes?
I have a problem where I need to calculate the sum of the probabilities of certain outcomes (events are A,B,C or D), but would like to do it in one formula.
Currently I have the following data:
...
- 101
0
votes
0
answers
62
views
Double infinite product $\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$
Question
Compute the products:
$\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$
$\prod_\limits{0<i<j<2020} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\...
- 1
0
votes
0
answers
29
views
Multiplication operations on sets - Which makes the distributive law hold?
Let $A,B,C$ be subsets of a ring $R$. Let's consider two arithmetic operations on sets.
We know that the sum of $A$ and $B$ is defined by $A+B=\left\{ a+b: a\in A,b\in B \right\}$.
For multiplication, ...
- 325
1
vote
1
answer
37
views
Neighborhood basis in product topology
Let $x = (x_1, x_2, \dots) \in \{0,1\}^{\mathbb{N}}$. Show that the sets $$B_n(x)=\{(y_1, y_2, \dots) \mid y_i = x_i \text{ for all } i =1,2,\dots,n\}$$ form a neighborhood base at $x$.
Let $O$ be a ...
- 1,384
15
votes
3
answers
306
views
Showing that $\prod_{k=1}^{n} \left( 3 + 2\cos\left(\frac{2\pi}{n+1}k\right) \right)$ is the square of a Fibonacci number
I was experimenting with products of the form
$$\prod_{k=1}^{n} \left( a + b\cos(ck) \right)$$
when I found that the expression
$$\prod_{k=1}^{n} \left( 3 + 2\cos\left(\frac{2\pi}{n+1}k\right) \right)$...
- 1,674