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

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5
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1answer
85 views

Is the product of all objects of a finite category an initial object?

If the product of all objects in a finite category exists, is it an initial object? I presume so, but I'm still learning this subject and I can't make a proof go through. Advice welcome. (Not a ...
1
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0answers
37 views

$(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$

Let $X_1$, $X_2$, and $X_3$ be spaces. (a) Prove that $(X_1 \times X_2) \times X_3$ is homeomorphic to $(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$ So, I think I ...
1
vote
1answer
78 views

How to minimize $a \times b$ where $a^b≥x$?

For example, if $x$ is 1 billion, the smallest possible $a \times b$ will be $3 \times 19 = 57$. This is because: $2^{30} >= 1000000000$ $2 \times 30 = 60 $ $3^{19} >= 1000000000$ $3 \times ...
1
vote
1answer
25 views

Is the nth root of a product of n terms used in place of the average anywhere?

In applied usage we typically take the average of values or terms which is done by summing them and dividing by the number of terms (for simple average): $$\sum_{i=1}^n \frac{a_i}{n}$$ It dawned on ...
0
votes
1answer
37 views

Double Product of a series

So in this proof (please don't ask about it, it's not important and it would take ages to explain) there's this step where they "switch" the values of the series of the double products in the ...
0
votes
1answer
28 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
0
votes
1answer
17 views

proving $(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem

$(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ this exercise is taken from Apostol's Calculus I (page 45) and it's supposed to be proved by using the binomial ...
0
votes
1answer
14 views

Counting zeros in product of numbers

This is surprising a simple asked question... How many zeros does the product $25^5$,$150^4$ and $2008^3$ end with? (A)5 (B)9 (C)10 (D)12 (E)13 The problem is,I am not allowed to use calculator ...
1
vote
1answer
37 views

Product of two vectors

Let $x, y \in \mathbb{R}^n$, when $x^T y = y^T x$ ?
0
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0answers
14 views

Approximate distribution of product of N normal i.i.d.?

Given $N>30$ i.i.d. $X\approx\mathcal{N}(\mu_X,\sigma_X^2)$, looking for: accurate closed form distribution approximation of $Y=\prod_{n=1}^{N}{X}$ asymptotic normal approximation of same ...
3
votes
4answers
52 views

Sum and product of integers conserving order

I have the feeling this is true, but can't prove it: $$\sum_n^An\lt\sum_n^Bn\implies\prod_n^An\lt\prod_n^Bn$$ Where $A\subset\mathbb N-\{0, 1\},B\subset\mathbb N-\{0, 1\}$ Example: ...
1
vote
1answer
19 views

Summing up decrementing geometric series?

Is there any easy way of summing up, $c,z \in R$ $z < 1, c < z $ $ k,n\in N$: $$\large\sum_{k=0}^{\lfloor\frac{z}{c}\rfloor}\prod_{n=0}^{k}(z-nc)^n$$ I'm searching for a formula to sum up ...
0
votes
1answer
49 views

Solving a Pi product.

How the value of this $P_k$ is calculated from the first equation. Thank you. $$k \geq m$$ $$P_k=P_0\prod_{i=0}^{m-1}\frac{\alpha}{(i+1)\mu}\prod_{j=m}^{k-1}\frac{\alpha}{m\mu}$$ ...
0
votes
2answers
36 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
2answers
40 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 ...
3
votes
2answers
100 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
0answers
33 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, ...
0
votes
1answer
29 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 = ...
0
votes
1answer
68 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
68 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
22 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
votes
1answer
551 views

Simplifying a product written in Capital Pi Notation

I'm having some trouble figuring out how to simplify Capital Pi Notation. What I tried was to expand the multiplication with various n and tried to find a pattern. Could someone point me in the ...
-2
votes
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?
2
votes
1answer
41 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
78 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
24 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
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0answers
36 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
57 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
votes
0answers
13 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
29 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!
0
votes
1answer
37 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 ...
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
70 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
47 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
39 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
40 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 ...
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0answers
39 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 ...
5
votes
3answers
143 views

Expressing $\prod_{k=1}^n \left( k - \frac{1}{2} \right)$ using the gamma function

I want to express $$\prod_{k=1}^n \left( k - \frac{1}{2} \right)$$ using the gamma function. I think this is equivalent to $\left(k-\frac{1}{2}\right)!$ so I set $a=k-1$ and then used the identity ...
6
votes
2answers
53 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
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?
1
vote
3answers
209 views

Why isn't every coproduct a product (and vice-versa)?

So I know that every coproduct is not a product, so I am misunderstanding some part of the definition of (co)products. Saying that $U$ is a coproduct (the disjoint union of $X_1$ and $X_2$ below) of ...
0
votes
1answer
54 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
4answers
120 views

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

Evaluate $$\prod_{k=2}^{2015} \left(1-\frac1{k^2}\right) = \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{2014^2}\right)\left(1-\frac{1}{2015^2}\right)$$
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 ...
2
votes
1answer
88 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 ...