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

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7
votes
1answer
797 views

Evaluation of a product of sines [duplicate]

Possible Duplicate: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I am looking for a closed form for this product of sines: \begin{equation} \sin ...
1
vote
1answer
23 views

Is the value of $c$ in $\prod_{i=1}^{n}\frac{p_i}{p_i-1}<e^c \cdot (\log p_n) \cdot(1+\frac{1}{\log_2p_n})$ known?

I Recently read this paper by Rosser and Schoenfeld (http://projecteuclid.org/download/pdf_1/euclid.ijm/1255631807) In Theorem 8, corollary 1, they state: $$\prod_{i=1}^{n}\frac{p_i}{p_i-1}<e^c ...
2
votes
2answers
42 views

Mathematical formalism for the “dot product” of three vectors

I know that the dot product of two vectors is the sum of element-wise multiplication. Using pseudo-MATLAB notation: (x,y) = sum(x.*y). I'm interested in ...
0
votes
1answer
10 views

Calculating better value products.

The special promotion tins of 300g cost 0.80$. The soup can also be bought in larger tin of 500g that cost 1.12$. Is it better value to buy the 500g tin or the special promotion tin? Show your ...
3
votes
1answer
77 views

Question about Meijer-G definition and identity

I'm trying to wrap my mind around computation involving the Meijer $G$ function, as defined here. (Edit: I'm actually using a somewhat mixed notation using the definitions from MathWorld and the ...
4
votes
2answers
35 views

How to get to this equality $\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1$?

How to get to this equality $$\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1?$$ I was studying the Euler Gamma function as it gave at the beginning of its history, and need to ...
0
votes
0answers
20 views

Partial sum of a product over an arbitrary sequence.

Below is an equation a friend showed me, but was unable to prove. After struggling with it for a bit I was unable to as well. After failing to show this for, say, N=2 Im pretty sure the equation is ...
5
votes
2answers
159 views

Simplify $\prod_{k=1}^5\tan\frac{k\pi}{11}$ and $\sum_{k=1}^5\tan^2\frac{k\pi}{11}$

My question is: If $\tan\frac{\pi}{11}\cdot \tan\frac{2\pi}{11}\cdot \tan\frac{3\pi}{11}\cdot \tan\frac{4\pi}{11}\cdot \tan\frac{5\pi}{11} = X$ and ...
3
votes
2answers
71 views

Prove that $\frac{1}{1999} < \prod_{i=1}^{999}{\frac{2i−1}{2i}} < \frac{1}{44}$

Prove that $$\dfrac{1}{1999} < \prod_{i=1}^{999}{\dfrac{2i−1}{2i}} < \dfrac{1}{44}$$ from the 1997 Canada National Olympiad. I have been able to prove the left half of the inequality using ...
3
votes
2answers
129 views

Matrix product notation

My lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means? Thanks in advance. Edit: ...
5
votes
2answers
145 views

Closed form for $(2^1-1)(2^2-1)…(2^k-1)$?

Is there closed form for $\prod_1^{i=k}(2^i-1)$ ? I found that it is the product of the terms of the following arithmetico-geometric sequence : $$\{u_1=1,u_{n+1}=2u_n+1\}$$ I found nothing with ...
2
votes
1answer
33 views

If $\ne: X \times X \to S$ is continuous, is X hausdorff?

The Sierpiński space is defined like so: $$S = (\{\top, \bot\}, \{\emptyset, \{\top\}, \{\top, \bot\}\})$$ (A nice way to visualize is to take [0, 1], and glue 0 on $\bot$ and (0,1] onto $\top$.) ...
3
votes
0answers
44 views

Polynomial products [duplicate]

This problem $$ \large \displaystyle\prod \limits^{14}_{k=1}\cos \left( \frac{k \pi }{15} \right) =\ ? $$ I solved it in this way $$ x = \displaystyle \prod \limits^{14}_{k=1}\cos \left( \frac{k\pi ...
0
votes
1answer
58 views

Difficult product problem $\prod \limits^{2014}_{k=1}\left( 1-\frac{1}{k^{2}} \right)$ [duplicate]

Evaluate the product $$\prod \limits^{2014}_{k=1}\left( 1-\frac{1}{k^{2}} \right)$$ Any help will appreciated!
3
votes
2answers
4k views

Does the product of two invertible matrix remain invertible?

If $A$ and $B$ are two invertible 5*5 matrices, does $B^{T}$$A$ remain invertible? I cannot find out is there any properties of invertible matrix to my question. Thank you!
18
votes
6answers
428 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
6
votes
2answers
106 views

Calculate $\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$

Calculate $$\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$$ I can only bound it as follows: $$\binom{n}{i}<\left(\dfrac{n\cdot e}{k}\right)^k$$ $$\sum_{i = ...
5
votes
2answers
251 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.
1
vote
1answer
65 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 ...
2
votes
0answers
28 views

Existence of formulae for sines/cosines of products of angles in terms of sines/cosines of original angles? [duplicate]

There was something that I was getting a little curious about. We know that there are the so-called compound-angle formulae for calculating sines and cosines of sums of angles in terms of those of the ...
1
vote
1answer
32 views

Evaluate products and sums

Did I evaluate the following terms correctly? Does the set notation in example b) allow me to chose the order of the terms? $$ a) \sum_{i=1}^6 ix^{i+1} = x^2+2x^3+3x^4+4x^5+5x^6+6x^7 \\ b) \prod_{i ...
0
votes
1answer
32 views

Can $\dfrac{b_0}{a_0} + \dfrac{b_1}{a_1} + \dfrac{b_2}{a_2} + \dfrac{b_3}{a_3} + … + \dfrac{b_n}{a_n}$ be represented as …

Is this correct? (Last step $\rightarrow$ After taking L.C.M.) $\large \dfrac{b_0}{a_0} + \dfrac{b_1}{a_1} + \dfrac{b_2}{a_2} + \dfrac{b_3}{a_3} + ... + \dfrac{b_n}{a_n} = \sum\limits_{k=0}^{n} ...
10
votes
3answers
253 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 ...
1
vote
1answer
98 views

Proving formulas with products of Fibonacci numbers

While digging through my old notes, I stumbled upon some formulas involving multiplication of Fibonacci numbers that I discovered about 7 years ago (being fascinated with Fibonacci numbers at the ...
0
votes
0answers
30 views

Product of n uniformly distributed RVs

Let $X_j \sim U(a,b)$. What is the PDF of $\prod_{j=1}^n X_j$? I have seen some with $X_j \sim U(0,1)$ but I was wondering what the general form of the solution is for any $a$ and $b$.
4
votes
2answers
51 views

Closed form expression for products

How can I find a closed form expression for products of the following form $$\prod_{k=1}^n (ak^2+bk+c)\space \text{?}$$
0
votes
1answer
8 views

Evaluate and simplify multiplication of exponents with base e; polar forms

$$2e^{(i×\pi/4)}×3e^{(i×\pi/6)}$$ How would I evaluate and simplify the above, and then express it in polar form? I understand $re^{i\theta} = r(\cos\theta+i\,\sin\theta)$. The question is to find ...
2
votes
2answers
66 views

Product of the first $N$ factorials

I'm trying to find a formula for the product of factorials: $$\prod _{n=1}^{N}n!=\; ?$$ Now using a kind of "brute force", I believe that I can prove that $$\prod _{n=1}^{N}n!=\prod ...
22
votes
3answers
4k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
1
vote
3answers
36 views

Product of Uniform Distribution and $\Gamma(2,1)$ Distribution

I ran into an old exercise but I seem to have messed up somehow. Can you tell me what went wrong? Let $U \sim \mathrm{Unif}(0,1)$ and $V \sim \Gamma(2,1)$ with $U,V$ independent. Show that $UV$ has ...
2
votes
1answer
25 views

independence of random objects when forming product spaces

Suppose we have two probability spaces $(\Omega_1, \mathscr{F}_1, \{\mathcal{F}^1_t\},\mathbb{P})$ and $(\Omega_2, \mathscr{F}_2, \{\mathcal{F}^2_t\},\mathbb{P}_2)$, if we take product space $$\Omega ...
3
votes
1answer
38 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...
4
votes
2answers
91 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)) = ...
1
vote
2answers
93 views

Product limit with exponentials

Find an explicit formula for the limit: $$\lim_{n \rightarrow \infty} n \prod_{k=2}^{n} (2 - e ^ {\frac 1 k})$$ I am not asking for convergence proof since I know the sequence is decreasing and ...
0
votes
2answers
44 views

Proof of associativity of polynomials product (infinite variables)

The product of polynomials in $R[X_i]_{i\in I}$ where $I$ is not necessarily finite is associative ($R$ commutative ring), but I can't find any detailed proof of this fact. Either it is left in ...
28
votes
4answers
836 views

Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
15
votes
2answers
304 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 ...
2
votes
3answers
488 views

Change from product to sum

We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : ...
2
votes
1answer
54 views

What is this (unusual) matrix/vector operation called?

A typographical error let to an unexpected (but, for me, potentially useful) result: $$ \left\{\begin{array} & a & b & c\\ d & e & f \\ g & h & i ...
2
votes
1answer
62 views

Show that $\prod _{i<j}(x_i-x_j)$ can be divided wihout remainder in $\prod_{i<j}(i-j)$ [duplicate]

Let $x_1,...,x_n$ be a natural numbers, show that $\prod _{i<j}(x_i-x_j)$ can be divided wihout remainder in $\prod_{i<j}(i-j)$ I know $\prod \left(x_i-x_j\right)$ is the result of ...
1
vote
4answers
71 views

Expansion of $x^n-y^n$

Studying polynomials I couldn't find a way to expand $x^n-y^n$ as a product of other polynomials. Now of course we know that $$x^4-y^4=(x^2+y^2)(x^2-y^2)=(x^2+y^2)(x+y)(x-y)$$ and I came up with this: ...
1
vote
0answers
30 views

Product of Dependent Bernoulli variables

Let $B_{i,n}$ with $i=1,...,n$ be the triangular Bernoulli array defined as $$ B_{i+1,n} = B_{i,n}\,R_{i+1,n}+\left(1-R_{i+1,n}\right)\,F_{i+1,n}, $$ where $R_{i,n}$ and $F_{i,n}$ are iid Bernoulli ...
2
votes
1answer
15 views

Product of Bernoulli variates

I am stuck with something that looks very simple but I am not able to find where I am wrong. Let $\xi_k$ with $k=1,...,n$ be $n$ iid Bernoulli random variables such that $$ ...
1
vote
0answers
19 views

The product of distribution taken over Unions

I have a probability problem as follow: $\mathbb{P}\big[\mathop{\arg\sup}_{x \in \bigcup_{i\in \{1,2\}} \Phi_{k,i} } \mathcal{f}(x )\geq y \big] = 1-\mathop{\prod}_{x \in \bigcup_{i\in \{1,2\}} ...
0
votes
2answers
47 views

How to do multiplication (capital pi) in WolframAlpha?

How do i ask this in WolframAlpha: $$\prod_{i=0}^{i=10} \sin{(i)}$$ I used $\text{product}(...)$ and $\text{multiply}(...)$ or even $\text{multiplication}(...)$ but they don't seem to work. I am ...
0
votes
1answer
36 views

What do you need to perform Karatsuba multiplication?

Karatsuba multiplication is usually defined in $\mathbb{N} \times \mathbb{N}$ and computes $$(aB^m+b)(cB^m+d)=acB^{2m} +[(a+b)(c+d)-ac-bd]B^m+bd$$ (where B is the base, usually 10) in only three ...
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
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 ...
1
vote
3answers
48 views

About $0!=1$ and $a^0=1$ as cases of empty product.

Some useful ''conventions'' as $0!=1$ or $a^0=1$ are particular cases of an empty product, i.e. a product between elements of the empty set. I know that such product is defined as a convention by: $$ ...
5
votes
2answers
560 views

Closed form expression for the product $\prod\limits_{k=1}^{n}\left(1 - \frac{1}{ak}\right)$

A simple method for evaluating a product is term cancellation. For example, the product $$\begin{align*} \prod_{k=2}^{n}\left(1-\frac{1}{k}\right)&=\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)\\ ...