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

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0
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1answer
38 views

$\prod_{n=1}^{\infty} (1+ (\frac{2\pi n}{\beta})^{-2} )^{-1} = \frac{\beta}{2 \sinh(\frac{\beta}{2})}$

\begin{align} \prod_{n=1}^{\infty} \left(1+ (\frac{2\pi n}{\beta})^{-2} \right)^{-1} = \frac{\beta}{2 \sinh(\frac{\beta}{2})} \end{align} I'd like to prove the following products. Can you give me ...
6
votes
2answers
54 views

Is there a geometric interpretation of the product integral?

Riemann's "way to the Integral" is loosely speaking the limit of sums of this kind \begin{equation} \sum_if(x_i)\Delta x_i \end{equation} Now, if we replace the sum with a product and the ...
0
votes
0answers
57 views

How to simplify sine function

Does anyone have an idea for simplifying this formula? $$f(x)=\prod\limits_{k=2}^{14}\sin(\frac{15x\pi}{k})$$ Or even more general case: $$f(x,y)=\prod\limits_{k=2}^{y-1}\sin(\frac{xy\pi}{k})$$ ...
2
votes
1answer
60 views

Product in category TOP(2)

Let TOP(2) be the category whose objects $(X,A)$ are pairs of topological spaces and whose morphisms $f:(X,A) \to (Y,B)$ are continuous maps $f:X\to Y$ such that $f(A) \subset B$. If I am not ...
2
votes
2answers
39 views

Pi product notation

The exact expression I've seen in a paper looks like this: $$\displaystyle \prod_{k<l}^L(x_k-x_l)$$ where $L$ is some natural number. What does the product actually look like when expanded out?
0
votes
1answer
55 views

Inverting a product

Can anyone explain why $$\prod^{0}_{n=5}\frac{1}{f(n)}=f(1)f(2)f(3)f(4)$$ in other words is there some relationship or identity for dealing with inverses in products.
0
votes
0answers
24 views

Computing product of lots of matrices?

I'm trying to compute the first column of $M$ where $$M=(A - x_1I)(A - x_2I)\cdots(A - x_rI)$$ where $A$ is in $R^{n \times n}$ and $x$ is a vector in $R^r$. Whatever way I think of it, it ...
0
votes
4answers
123 views

Evaluate $(1-\frac1{2^2})(1-\frac1{3^2})\ldots(1-\frac1{2015^2})$ [closed]

Evaluate $$\prod_{k=2}^{2015} \left(1-\frac1{k^2}\right) = \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{2014^2}\right)\left(1-\frac{1}{2015^2}\right)$$
2
votes
1answer
94 views

Sum over subsets of $\{1,2,\ldots,n\}$ of terms involving a product over that subset

I'm attempting to perform a sum, using products, using all possible combinations, in a function. How would I go about doing this? (I really need to find something that works.) For example, say I ...
1
vote
2answers
75 views

Normable topology determined by its restriction to a finite number of factors?

Is it generally true that all norms $\|\cdot\|$ on a finite product of normed spaces $E_1\times\dots\times E_n$ with $\|(0,\dots,0,x,0,\dots,0)\|=\|x\|_i$ where $\|\cdot\|_i$ denotes the norm on $E_i$ ...
1
vote
1answer
16 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 ...
2
votes
0answers
52 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}-T_{N_j})(x)$ I'm trying to define a polynomial $p_{k,m}(x)$ like ...
1
vote
2answers
41 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
69 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
64 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
73 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
292 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
28 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
65 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
17 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
131 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
111 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
20 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
57 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
438 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
123 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
68 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
52 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
51 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
76 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
53 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
44 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
96 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
104 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
327 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
65 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
72 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
88 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
39 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
38 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
101 views

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

Related. Show that if $x$ is large enough,$$\prod_{\substack{p<x \\ p \ \text{prime}}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this ...
7
votes
2answers
425 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
138 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
100 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
40 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
36 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 $S$ ...
3
votes
1answer
116 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
121 views

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

For example $\prod_{2 \le j < 1} 2^j= 1.$ How does that happen?
1
vote
0answers
98 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
98 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 ...