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

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-1
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1answer
18 views

Generalized Holder Inequality

Let $a_i \in \mathbb R^n$ with $a_i = (a_{i}^j)_{j = 1 ... n} = (a_{i}^1, ... ,a_{i}^n)$ for $i = 1, ... , k$ and let $p_1,...,p_k \in \mathbb R_{>1}$ with $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
0
votes
2answers
37 views

Factorial inequality $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n$ using induction

I want to show $2!\,4!\,6!\cdots (2n)!\geq\left((n+1)!\right)^n.$ Assume that it holds for some positive integer $k\geq 1$ and we will prove, $2!\,4!\,6!\cdots (2k+2)!\geq\left((k+2)!\right)^{k+1}$. ...
5
votes
2answers
91 views

Study the convergence of $\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$

Can you help me to study the convergence of the following series: $$\sum_{n=1}^{\infty}{\prod_{k=1}^{n}{\sin (k)}}$$ Thanks.
7
votes
4answers
132 views

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to? As far as I can tell, this has no closed-form solution (not saying much, I don't know much math), but a friend of mine swears he saw a ...
1
vote
0answers
63 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
6
votes
0answers
190 views

Two (strictly related) proofs by induction of inequalities.

Predictably, I'm stuck with the inductive steps. Let $p_m$ be the largest prime factor of $a_n$ and set $\lim_{n\to \infty}\frac{\log a_n}{p_m}=1$. Suppose also this ratio converges to $1$ faster than ...
2
votes
1answer
66 views

Improvements of Dusart's lower bound for $ e^\gamma \log x \prod_{p\le x} \frac{p-1}{p}$.

Let $\gamma$ be the Euler-Mascheroni constant. In this paper (Theorem 6.12) it is proved that for $x\ge 2793$, $$ e^\gamma \log x \prod_{p\le x} \frac{p-1}{p}> 1-\frac{1}{5 \left(\log ...
3
votes
1answer
42 views

Is the term “telescoping product” well known?

I know that "telescoping series" (or sum) is well known. But I can't find many reliable references to the term "telescoping product". It would be one of the following: $x_i = \dfrac{y_i}{y_{i+1}}$: ...
7
votes
2answers
164 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
2
votes
0answers
34 views
2
votes
1answer
127 views

Is $\frac{1}{e^\gamma\log x} \prod\limits_{p < x,p\,\text{prime}} \frac{p}{p-1}<1+ \prod\limits_{p<x,p\,\text{prime}}\frac{1}{p^{n+1}-1}?$

Let $n$ be an initially arbitrarily large variable, but always decreasing (and more specifically non-increasing) to exactly $1$ when $p$ is the largest prime in the product. Then, denoting with ...
1
vote
0answers
36 views

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$?

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$? If so, where can I find the equivalent of a Wikipedia entry?
5
votes
1answer
84 views

Being $N$ a constant $>0$, show $\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that if $x$ is large enough,$$\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this paper might be ...
0
votes
2answers
86 views

What's the name of this strange inequality?

There is an inequality: $$\sqrt[n]{\prod_{i = 1}^{n}{(a_i+b_i)}} \geq \sqrt[n]{\prod_{i = 1}^{n}{a_i}} + \sqrt[n]{\prod_{i = 1}^{n}{b_i}}$$ which I even don't know its name. I'd like to have an ask ...
8
votes
1answer
208 views

Olympiad problem: Erdos-Selfridge

The following problem is a special case of Erdos-Selfridge theorem: http://projecteuclid.org/euclid.ijm/1256050816 Problem: Prove that for any positive integer $n$, the product $(n+1)(n+2)...(n+10)$ ...
0
votes
1answer
185 views

Weak direct product

I am just reading the book "Algebra" by Hungerford and on one page it says that if $G_i$ is a family of groups $\forall i\in I$ then $\prod_{i\in I}^{w}G_i\unlhd\prod_{i\in I}G_i$ where ...
3
votes
1answer
19 views

Product rule for Hessian matrix

Let $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}$. Is there a general formula for the Hessian matrix of their product? That is, what is $H(f(x) g(x))$, where $H(f(x)) = ...
20
votes
2answers
3k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
3
votes
0answers
39 views

How I can calculate this product

How I can calculate this product: $$s=∏_{j=1}^{p-3}\sinh^2[2^{j-1}\cosh^{-1}( 2)]$$ for a natural number $p>3$.
0
votes
2answers
28 views

$S$ nonempty finite subset of group $G$ for which $SS=S$. $S$ is subgroup.

Let $S\subset G$, $S$ finite and nonempty, $G$ group. Suppose additionally that $$SS=\{s_1 s_2: s_1\in S, s_2 \in S\}=S.$$ How can I prove that $S$ is a subgroup of $G$? Does this hold for $S$ ...
2
votes
1answer
53 views

Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
2
votes
1answer
44 views

Product of ergodic transformations

I'm asked to give an example, that the product of two ergodic systems is not ergodic in general. I know that for $X_1=X_2=(S^1,B,m,R_a)$ (the irrational rotation on the unit circle with Lebesgue ...
2
votes
0answers
73 views

Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true. (in other word for every question ...
6
votes
9answers
76 views

Why is empty product defined to be $1$? [duplicate]

For example $\prod_{2 \le j < 1} 2^j= 1.$ How does that happen?
5
votes
2answers
102 views

Characterize the type of sequence that satisfies $\prod (1-a_i) \leq c$

Consider a product $\prod_{i=1}^{n} (1-a_i)$ where $n\leq \infty$ and $a_i\in [0,1)$ for all $i$. I'm hoping to see if there exist conditions on the sequence $\{a_i\}$ so that $$\prod_{i=1}^{n} ...
2
votes
2answers
90 views

Idea for primality testing based on a trigonometric product

This is an idea that I had about 3 months ago. I tried some college professors, they didn't care. I tried to solve, but with no luck. I ask for help to find the closed form of the following product ...
9
votes
1answer
134 views

How to compute the following integral in $n$ variables?

How can the following integral be calculated: $$ I_n=\int_0^1\int_0^1\cdots\int_0^1\frac{\prod_{k=1}^{n}\left(\frac{1-x_k}{1+x_k}\right)}{1-\prod_{k=1}^{n}x_k}dx_1\cdots dx_{n-1}dx_n $$ There should ...
0
votes
1answer
28 views

Recurrence relation for the coefficients of the polynomial $p_n(x) = \prod_{i=0}^{n-1}(x-i)$

Let's consider the polynomials $$ p_n(x) = \prod_{i=0}^{n-1}(x-i)=\sum_{i=1}^{n} a_{n,i}x^i$$. for all $n \in \mathbb{N}$. If $n=1$, then $p_1(x) = x$ and $a_{1,1} = 1$. Since I know that: ...
0
votes
0answers
66 views

Help in writing a nasty expression in nice closed form

This question is abouting re-writing a product in nice closed form. I have the following $$f(v_1) = \left(\sum_{i=1}^K \pi \lambda_i \delta_1 v_1^{\delta_1-1} P_i^{\delta_1} e^{-\beta_i ...
1
vote
1answer
43 views

Values of $x$ for convergence

I was posed this problem, it took me a while to solve it – but, I did nevertheless. I shall pose it for all of you, too. In my opinion it is a great exercise. For what values of $x$ is the series ...
0
votes
1answer
31 views

Theorem? For any sets A, B, C, and D, if A x B is a subset of C x D then A is a subset of C and B is a subset D.

  Is the following proof correct? If so, what proof strategies does it use? If not, can it be fixed? Is the theorem correct?   Proof. Suppose A x B is a subset of C x D. Let a be an arbitrary element ...
1
vote
1answer
55 views

The limit of products of the form $(n^3-1)/(n^3+1)$

Calculate $$\lim_{n \to \infty} \frac{2^3-1}{2^3+1}\times \frac{3^3-1}{3^3+1}\times \cdots \times\frac{n^3-1}{n^3+1}$$ No idea how to even start.
1
vote
2answers
72 views

being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$?

Let's say that I have a vector $\mathbf{w}$. How can I calculate the derivative in the following expression? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$ Update: found these ...
3
votes
1answer
34 views

How to calculate the product of a set

How can you calculate the product of a set $A$, denoted by $\Pi A$ and defined by $\forall z \in \Pi A(z \subseteq \bigcup A \wedge \forall y \in A (\exists x (z \cap y = \lbrace x \rbrace))) $ ...
3
votes
1answer
96 views

Resemblance between product and homotopy

The notion of product $X\times X$ for an object $X$ of a category $C$ resembles the notion of homotopy between two continuous functions. Indeed the relevant diagrams look the same: ...
1
vote
2answers
26 views

relationship between multiplication and correlation

is there a deep interpretation of multiplication as correlation? is this in some sense the most fundamental way to "combine" objects (eg numbers) into products? my reasons for asking are that the ...
1
vote
0answers
53 views

Simplify the product of two sums

How can I simplify the following product of two sums: $$ \biggl(\, \sum ^{n}_{k=0}a_{k}\biggr) \biggl(\, \sum ^{n}_{k=0}\dfrac {1}{a_{k}}\biggr) $$
4
votes
2answers
53 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
0
votes
1answer
35 views

Using induction to prove that $ \prod_{i=1}^{n} (1+a_{i}) \geq 1 + \sum_{i=1}^{n}a_{i} $ [closed]

I started a course in my university and I am having trouble with answering this question: Prove using Mathematical induction, for every real, non-negative 'n' number $$(a_{i}\geq 0)$$ the ...
0
votes
0answers
24 views

Product of dot products of two vectors

I have a product of innerproduct/dot product of two vectors. $ \langle u_i,v_j \rangle\cdot\langle x_i,y_j\rangle$. Is there any transformation/decomposition such that I can combine $u_i$ with $x_i$ ...
-1
votes
3answers
28 views

Product of inner products

Is product of innerproduct again a inner product of two vectors? For example - Is $ (< u,v >)(< x,y >) = < m,n > $? And if yes is m and n unique and how do we calculate those?
0
votes
1answer
58 views

Definition of a coproduct and its universal property - connection?

I have a problem connecting the definition of a coproduct with its often mentionend universal property. Let's start with the definition (just for two objects): Let $A_1$ and $A_2$ be objects of a ...
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0answers
39 views

Can I do the following when solving my integration??

I appreciate any feedback for my question. I have an integration as follows $$\int_{-\pi}^{\pi}\frac{1}{2\pi} \prod_i \frac{1}{1+ x_ig(\theta)} d\theta $$ I have that $g(\theta)$ is the defined as ...
1
vote
3answers
25 views

summation and product of sin and cos

I wonder how to find summation for $\displaystyle \sum_{k=0}^{n-1}(\cos{\frac{2\pi k}{n}+i \sin\frac{2\pi k}{n}})$ and the same for product $\displaystyle \prod_{k=0}^{n-1}(cos{\frac{2\pi k}{n}+i ...
1
vote
1answer
80 views

Product of roots of unity

Does somebody have a nice proof of the following? $$\prod_{m=1}^{n-1} \frac{e^{2\pi i k m/n} - 1}{e^{2 \pi i m / n} - 1} = \begin{cases} 1 & \text{ if $\gcd(k, n) = 1$} \\ 0 & ...
1
vote
0answers
25 views

Product notation $\prod$ when product does not commute [duplicate]

This is kind of a dubious question, but is the product notation $\prod$ often used in noncommutative rings? For example, if $M_i$ are matrices, I guess the common definition of $\prod$ is $$\prod_i ...
1
vote
3answers
200 views

Product and Square Root Proof

Let $a_1$ and $a_2$ be positive integers and let $m = a_1 a_2$. Prove that at least one of $a_1$ or $a_2$ is at least $\sqrt m$. Disclosure: This is for a homework question, though the question is ...
0
votes
0answers
26 views

writing sum as a product and vice versa.

$\Pi = k$ from k = 1 to n Can you write this in form of sigma? So that you can evaluate it as a sum? Also, are there any shorthand formula to evaluate a product like there are for summations? ...
2
votes
0answers
22 views

Vectorial product analog operation in 4+ dimensions?

I am thinking about a such operation. Which it need to have: It needs to be $\mathbb{R}^n\times{\mathbb{R}^n}\rightarrow\mathbb{R}^n$ The result needs to be perpendicular to the arguments (thus, ...
1
vote
2answers
32 views

Proofs of identity for product of binomial coefficients

While verifying my answer to another question, I came across a problem of binomial coefficients: Does $\hspace{.2cm}\displaystyle \prod_{k=1}^{n-1}\binom{n-1}{k}=\prod_{k=1}^{n-1}k^{2k-n}$ for all ...