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

learn more… | top users | synonyms

1
vote
1answer
12 views

Reference request for a special product

I have the product $$\prod_{k=0}^n (1+a_k)$$ Does this product have a special name under which I can find some of its properties? I appreciate any reference for this product. Note: Because of the ...
0
votes
0answers
12 views

Cleaning Up Messy Product Notation

Suppose I have the following: Let $N_1<...<N_m$. Let $T_{N_k}(x)=\sum_{i=0}^{N_k}{\frac{x^i}{i!}},$ $ t(i,j,x)=T_{N_i}(x)-T_{N_j}(x)$ I'm trying to define a polynomial $p_k(x)$ like this: ...
1
vote
2answers
35 views

Does the dot product angle formula work for $\Bbb{R}^n$?

Whenever I have seen this formula discussed \begin{equation} \textbf{A} \cdot \textbf{B} = \|\textbf{A} \| \|\textbf{B} \| \cos\theta \end{equation} I have always seen it using vectors in ...
2
votes
3answers
62 views

How to prove the identity $\prod_{n=1}^{\infty} (1-q^{2n-1}) (1+q^{n}) =1 $ for $|q|<1$?

Eulers product identity is as follows \begin{align} \prod_{n=1}^{\infty} (1-q^{2n-1}) (1+q^{n}) =1 \end{align} How one can explicitly prove this identity? Note here $q$ deonotes a complex number ...
0
votes
3answers
36 views

Product of random independent variables

What are the properties of the product of random variables? The book I have on probability and statistics only comments on their sum properties. I know that when you sum random independent variables ...
1
vote
1answer
40 views

Expected value of a product of n random variables

I am currently dealing with an expression of the form $\operatorname E[\Pi_{i=1}^n X_i]$, where $\operatorname E$ represents the expectation value and $X_i$ is an arbitrary random variable. ...
8
votes
1answer
214 views

Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose ...
1
vote
2answers
27 views

computation $ \prod_{n=1}^{\infty}(1+q^{2n})(1+q^{2(n-1)}) = 2 \prod_{n=1}^{\infty} (1+q^{2n})^2 $

I want to compute the following identity $ \prod_{n=1}^{\infty}(1+q^{2n})(1+q^{2(n-1)}) = 2 \prod_{n=1}^{\infty} (1+q^{2n})^2 = \frac{1}{2} \prod_{n=1}^{\infty} (1+q^{2(n-1)})^2 $ Can anyone gives ...
1
vote
2answers
50 views

How to compute $\prod_{n=-\infty}^{\infty}(n+a) = a \prod_{n=1}^{\infty} (-n^2)(1- \frac{a^2}{n^2}) = 2 i \sin(\pi a) $

I want to compute the following identity $ \prod_{n=-\infty}^{\infty}(n+a) = a \prod_{n=1}^{\infty} (-n^2)(1- \frac{a^2}{n^2}) = 2 i \sin(\pi a) $ It seems strange this identity holds to me. Can ...
0
votes
1answer
16 views

Subsets Of A Set Product

I was asked the following question: Let $A={1,2,3}$ and $B={4,5}$. How many subsets does the set $A\times B$ contain of size at most $4$? My understanding of the outer product $A\times B$ is ...
4
votes
2answers
119 views

Closed form for $ \prod_{k=1}^n (a+k^2) $

I have come across the following product: $$ \prod_{k=1}^n (a+k^2) $$ where $a$ is a positive constant. Could anyone suggest a closed form for this product? I need to approximate this for large $n$, ...
3
votes
0answers
105 views

Simplification of recursive polynomials

Suppose I have known polynomials $p_i, i=0\ldots k-1$. I have the following horrific looking recursive equation: ...
0
votes
0answers
16 views

Estimates for a Mertens-type Product.

The first corollary of Theorem 8 of this paper by Rosser and Schoenfeld states that $$\prod_{p\leq x}\left(\frac{p}{p-1}\right)<e^{\gamma}(\log x)\left(1+\frac{1}{\log^2 x}\right)$$ for all $x\geq ...
1
vote
2answers
53 views

Limit $\lim_{k\to\infty}\prod_{m=2^k}^{m=2^{k+1}} {\frac{2m}{2m-1}}$

Does $$\lim_{k\to\infty}\prod_{m=2^k}^{m=2^{k+1}} {\frac{2m}{2m-1}}$$ exist? Wolfram alpha gives numbers around 1.4
10
votes
1answer
424 views

New Year Combinatorics

In the spirit of the festive period and in appreciation of the encouraging response to my X'mas Combinatorics problem posted recently, here's one for the New Year! Express the following as a ...
3
votes
1answer
119 views

Proving the inequality $ \left|\prod_{i=0}^n \left(x - \frac{i}{n}\right)\right| \le \frac{n!}{4n^{n+1}}$

Let $n \in \Bbb{N}$ and $x \in [0,1]$ prove $$ \left|\prod_{i=0}^n \left(x - \frac{i}{n}\right)\right| \le \frac{n!}{4n^{n+1}}$$ I manage to show that $\left| (x-\frac{n-1}{n})(x-\frac{n}{n})\right| ...
0
votes
2answers
63 views

Solve $p_4(x) = x^4 −(2m + 4)x^2 + (m−2)^2 $such that $p_4$ is a product of two non-constant integer-coeficient polynomials

I'm having trouble getting the starting idea for a problem I've been presented with: I need to find values for m (integer) such that the following polynomial $p_4(x) = x^4 −(2m + 4)x^2 + ...
1
vote
1answer
46 views

An inequality for the product $\prod_{k=2}^{n}\cos\frac{\pi }{2^{k}}$

I have this hint from old question of mine if someone could help me to understand it Sequence $0\leq a_{n}-l\leq \dfrac{\pi^{2} }{2^{2n+1}}$ for the first question it's easy to see that a_n is ...
1
vote
1answer
45 views

Preferred way to write elements of the direct sum of vector spaces

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We ...
1
vote
0answers
57 views

How to prove taht a product of two complete residue system is not a complete residue system?

Claim. Let $n$ be a natural number and $A=\{0,1,2,3,\cdots,n-1\}$ be a complete set of residues modulo $n$. Let $\sigma$ be a permutation of $A$. Show that the set $C=\{\sigma(i)i:i\in A\}$ is not a ...
1
vote
1answer
52 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
-1
votes
1answer
28 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 $\frac1{p_1}+ ... + \frac1{p_k} = 1$ ...
1
vote
3answers
86 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
100 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.
8
votes
0answers
300 views

Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
3
votes
1answer
45 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: $x_i = \dfrac{y_i}{y_{i+1}}$: ...
1
vote
0answers
67 views

Summation verification

I have a particular polynomial $$ 1-10x+35x^2-50x^3 $$ Which can be written nicely as $$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ ...
2
votes
1answer
83 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 ...
2
votes
0answers
35 views

How the second form of following equation is derived form first form (i.e. given first line, what are the steps involved in writing second line

How the second form of following equation is derived form first form (i.e. what are the steps involved in writing second line)
1
vote
0answers
37 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
88 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 ...
7
votes
2answers
185 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
1answer
135 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 ...
0
votes
2answers
95 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 ...
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
29 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
61 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 ...
6
votes
9answers
82 views

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

For example $\prod_{2 \le j < 1} 2^j= 1.$ How does that happen?
2
votes
0answers
82 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 ...
2
votes
2answers
93 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 ...
5
votes
2answers
104 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} ...
0
votes
1answer
30 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
1answer
60 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
56 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.
9
votes
1answer
137 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 ...
3
votes
1answer
35 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))) $ ...
2
votes
1answer
51 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 ...
1
vote
2answers
30 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
62 views

Simplify the product of two sums

How can I simplify the following product of two sums: $$ \biggl(\, \sum ^{n}_{k=0}a_{k}\biggr) \biggl(\, \sum ^{n}_{k=0}\dfrac {1}{a_{k}}\biggr) $$
3
votes
1answer
28 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)) = ...