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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2answers
33 views

Prove $n!\mid\prod_{k=i}^{i+n-1}k$

I have no idea how to prove this, I haven't yet learned much about this kind of product. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
1
vote
0answers
65 views

How to determine if a number is product of exactly two primes? [on hold]

Suppose $X = p_1p_2$ where $p_1$ and $p_2$ are (not necessarily consecutive) prime numbers. Given $X$, what are the different ways to prove or determine that $X$ is a product of two primes? We ...
1
vote
2answers
34 views

non-cartesian set product?

Foremost, this question is asked from a point of a computer scientist undergrad, so please don't nag me for inconsistent notation, or lack of proper vocabulary. Is there a concept in mathematics for ...
0
votes
0answers
28 views

Product-rule for Jacobian calculation, i.e. $\frac{d}{dx}(Ay)$ where A is a matrix and y a vector and both depend on x

I'm trying to understand a paper in which the author constructs sensitivity matrices in the process of linearizing an equation. I figured that the sensitivity matrix has to be a Jacobian Matrix, ...
3
votes
2answers
98 views

Product of repeated cosec.

$$P = \prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$$ I realize that there must be some sort of trick in this. $$P = \csc^2(1)\csc^2(3).....\csc^2(89) = \frac{1}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$ I ...
0
votes
1answer
65 views

Infinite product converges to meromorphic function

How do you show that $\frac{1}{z}\prod_{n=1}^\infty \frac{n}{z+n}(\frac{n+1}{n})^z$ is meromorphic? Any hints would be helpful, I'm having trouble bounding the functions and their logarithms. This is ...
4
votes
1answer
57 views

Sum and Product Puzzle and Prime Factors

Suppose we have two number $X$ and $Y,$ such that $1 < X < Y < 100,$ and $X + Y ≤ 100.$ Sue is given $S = X + Y$ and Pete is given $P = XY.$ They then have the following conversation: ...
2
votes
1answer
21 views

Measurable functions on product space

Let $(\Omega, \mathcal{H}), (E, \mathcal{E})$ and $(F,\mathcal{F})$ be measurable spaces. Let $(E \times F, \mathcal{E} \otimes \mathcal{F})$ be a product space. Define the following three functions: ...
-2
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2answers
52 views

how do I prove this matrix result? [closed]

How do I prove that if A and B are lower triangular matrices, then AB is also a lower triangular?
0
votes
1answer
26 views

Building matrix expressions for product of sum, isolating vector of constants

This identity to build the matrix expression for the expression below is pretty straightforward: $$ \left.\sum\limits_{j=1}^M \left( a_j \cdot f_{i,j} \right) \;\right|_{i=1}^N = ...
2
votes
1answer
40 views

Writing an expression as a product of products

I am currently dealing with the following expression: $$\left(\prod_{i=1}^{N-1}(\lambda_N-\lambda_i)\right)\left(\prod_{i=1}^{N-2}(\lambda_{N-1}-\lambda_i)\right)\cdots (\lambda_2-\lambda_1)$$ Is ...
1
vote
0answers
21 views

Continuity of Product Topology [duplicate]

Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a ...
0
votes
3answers
74 views

Fundamental theorem of algebra simple proof for rewriting with roots

My question is very basic, as I do not understand the concept of rewriting a (complex) polynomial into a product of terms using the roots of the polynomial. I have encountered the fundamental theorem ...
0
votes
1answer
20 views

How do I prove the following statement about the complement of a cartesian product?

How do I prove that this statement is true? $$(A\times B)^C=\left(A^C\times B\right)\cup\left(A\times B^C\right)\cup\left(A^C\times B^C\right)$$
2
votes
1answer
23 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
0
votes
0answers
33 views

Multiple sums or products in wolfram-alpha

How can I compute something like $$\prod_i^n \prod_j^m ij$$ in wolframalpha? (for finite n and m) I have tried a great number of combinations that have only resulted in failure.
2
votes
0answers
51 views

Solving a question by using special products (Students debate to Teacher)

So today,we got back our exam papers,and we found a question marked wrongly and teacher said that it is wrong.We all students do NOT believe this.So here is what happened. Before reading the next ...
0
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0answers
12 views

Multiply Vector and Matrix of Different Dimensions(Kronecker Product)

Suppose I have a vector ${\bf v} = (p_1,p_2, p_3, p_4, p_5, p_6, p_7, p_8, p_9)$, and I have a matrix of ${\bf M} = \left( \begin{array}{ccc} \lambda & -\lambda & 0 \\ 0 & \lambda ...
1
vote
0answers
28 views

Integral over a product

How the following integral can be computed: $I = \int_0^1 (x-a_1)^{b_1}(x-a_2)^{b_2}...(x-a_n)^{b_n} dx$? Here, $a_i,b_i$ are real numbers and $n$ is a natural number. Are there any techniques for ...
1
vote
2answers
82 views

How to prove $\underset{p\leq n}{\prod}p^{\frac{1}{p-1}}\leq 2n$?

I know that ($p$ prime) (1) $$\underset{p\leq n}{\prod}p^{\frac{1}{p-1}}\sim n$$ Is there a way to prove (2) $$\underset{p\leq n}{\prod}p^{\frac{1}{p-1}}< 2n$$ ? Thanks!
1
vote
1answer
38 views

What's the approximation for $\prod_{p\leq n^2} p^{2n}$?

I have 2 questions ($p$ prime): 1) I know that $$\underset{p\leq n}{\prod}p^{\frac{1}{p-1}}\sim n$$ Does that mean $$\underset{p\leq n^2}{\prod}p^{\frac{1}{p-1}}\sim n^2$$? 2) What's the ...
4
votes
1answer
69 views

Can $\prod_{i=1}^{\pi(n)} p_i^{\frac{1}{p_i-1}}$ be calculated?

Is there a way to calculate this Product as a function of $n$? $$\prod_{i=1}^{\pi(n)} p_i^{\frac{1}{p_i-1}}$$ where $p_i$ is the $i^{\text{th}}$ prime number, and $\pi(n)$ is the Prime-counting ...
2
votes
1answer
46 views

Is this expression bounded?

I wonder: is $$ \left( 1 + \frac{n}{a} \right)^{-a} \prod_{k = 1}^n \left( 1 + \frac{a}{k} \right) $$ uniformly bounded in $n \in \mathbb{N}$ and $0 < a \leq n$? Following Jack's answer, I have ...
0
votes
0answers
37 views

Topological Equivalence of Product Metric Spaces

Suppose that the metric space $(X_i,d_i)$ is topologically equivalent to $(Y_i,d'_i)$ for $i=1,2, \cdots , n$. Show that the product metric spaces $X = \prod_{i=1}^nX_i$ and $Y= \prod_{i=1}^nY_i$ are ...
2
votes
2answers
39 views

Is the product of atomic algebras necessarily atomic?

According to Terrance Tao's Measure Theory book, a boolean algebra $\mathcal{B}$ on a set $X$ is atomic, if there exist disjoint sets $(A_\alpha)_{\alpha \in I}$ which we refer to as atoms, such that ...
1
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0answers
35 views

Is the product of two discrete $\sigma$-algebras necessarily discrete?

I know that the answer to this question is negative, since proving the opposite is an exercise in Terrance Tao's Measure Theory book. However, it doesn't make sense to me. In another part of the same ...
0
votes
1answer
36 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 ...
0
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0answers
46 views

Working on $\prod_{k=1}^{n}\frac{c^k+1}{c^k-1}$

Let $n\in\Bbb{N},c>1$. I have recently encountered the following product in my work : $P_n(c)=\displaystyle\prod_{k=1}^{n}\frac{c^k+1}{c^k-1}$ What is known about this product ? I can't find any ...
6
votes
2answers
51 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
56 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
36 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
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1answer
52 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
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0answers
23 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
119 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
87 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
74 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
14 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
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0answers
51 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
39 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
66 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
45 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
55 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
276 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
57 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
123 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
109 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 ...