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

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0answers
48 views

Sum of products of K numbers taken from N numbers in closed form

Let's say i have 5 numbers, $A,B,C,D,E$. I want to know the sum of all the possible products of some or all of these numbers each taken at most once. Instead of a lot of multiplications and additions ...
0
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0answers
56 views

Solving a ratio of summation

I have to solve the equation \begin{equation*} y = \frac{\sum_{j=1}^m a_j x'_j}{\sum_{j=1}^m a_j x_j} \end{equation*} We have $\sum_{j=1}^m a_j \frac{x'_j}{x_j} = 1$ and $\sum_{j=1}^m a_j = 1$ ...
0
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0answers
51 views

Optimizing the trace of a matrix product

I have a problem where I have a NxT matrix P (lets just assume full rank for now, where N>>T) and a TxN inclusion matrix S. Each column of S must contain exactly one 1 and the rest 0's i.e. 1_T*S = 1, ...
5
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2answers
54 views

A trigonometric product

I have to prove: $$\prod_{i=1}^6 \left(2\cos\left(\frac{2^{i}\pi}{13}\right)-1\right)=1$$ I really have no idea about starting with this one. With the help of Wolfram Alpha, I noticed that: ...
24
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11answers
2k views

Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition? For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b ...
0
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1answer
35 views

Link between two products

Could someone help me to solve this problem : Let's denote by $A_i$ the following product, $$ A_i = \prod_{\substack{k=1 \\ k\neq i}}^n (a_k - a_i) $$ Is there any link or simple formula between ...
10
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3answers
254 views

What is the probability that the product of $20$ random numbers between $1$ and $2$ is greater than $10000$?

Twenty random real numbers $a_1,a_2,\dots,a_{20}$ are chosen such that $1\le a_i \le 2$. What is the probability that their product is greater than $10000$? (By random, I mean each real number in the ...
4
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2answers
78 views

If $\prod\limits_{k=0}^5(5^{2^k}+6^{2^k})=6^x-5^y$, what is the value of $x-y$?

I think this might be a contest math question, so I'm tagging it as such. I don't know how to do something like this by hand (or if it's even possible, though I would presume it is if it's from a ...
1
vote
0answers
49 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to ...
5
votes
1answer
109 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 ...
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vote
1answer
32 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
51 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 ...
1
vote
1answer
137 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: $2^{30} \ge 1000000000$ $2 \times 30 = 60 $ $3^{19} \ge 1000000000$ $3 \times 19 ...
0
votes
1answer
34 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 ...
1
vote
1answer
28 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 ...
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1answer
18 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 ...
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1answer
39 views

Product of two vectors

Let $x, y \in \mathbb{R}^n$, when $x^T y = y^T x$ ?
0
votes
0answers
19 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 ...
1
vote
1answer
21 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
56 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
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2answers
39 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. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
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2answers
48 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 ...
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0answers
48 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
110 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
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1answer
82 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
102 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
49 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: ...
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2answers
58 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?
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1answer
45 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
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 ...
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3answers
125 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
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1answer
22 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
29 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?
3
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0answers
108 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 ...
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0answers
27 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
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0answers
38 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
83 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
41 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
79 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
49 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 ...
2
votes
2answers
45 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
59 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
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1answer
39 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
64 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 ...
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0answers
59 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
66 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
44 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?
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1answer
57 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
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
124 views

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

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