For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.
3
votes
0answers
55 views
Distribution of Digit Products
A digit product $P(n)$ of a natural number $n$ is given by the product of its decimal digits. For example:
$$P(1234) = 24,\;\;\; P(24) = 8,\;\;\; P(8) = 8$$
$$1\times2\times3\times4 = 24, \;\;\; ...
2
votes
2answers
38 views
Simplifying a Product of Summations
I have, for a fixed and positive even integer $n$, the following product of summations:
$\left ( \sum_{i = n-1}^{n-1}i \right )\cdot \left ( \sum_{i = n-3}^{n-1} i \right )\cdot \left ( \sum_{i = ...
1
vote
0answers
31 views
Identity involving a recursive product
Here is yet another problem related to plane partitions. Given the recursive formula
$$
\begin{align*}
F(0)&=1,\\
F(r)&=\prod_{i=1}^r\frac{i+2r-1}{2i+r-2}F(r-1).
\end{align*}
$$
How can we ...
4
votes
4answers
86 views
Big Greeks and commutation
Does a sum or product symbol, $\Sigma$ or $\Pi$, imply an ordering?
Clearly if $\mathbf{x}_i$ is a matrix then:
$$\prod_{i=0}^{n} \mathbf{x}_i$$
depends on the order of the multiplication. But, ...
3
votes
2answers
56 views
How to define this pattern as $f(n)$
Given a binary table with n bits as follows:
$$\begin{array}{cccc|l}
2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ ...
3
votes
1answer
44 views
Show that the following product equals 1 (involves trig)
How can I show that:
$$\prod_{k=1}^{n}\left ( 1+2\cos\frac{2\pi .3^{k}}{3^{n}+1} \right )=1$$
Could you please explain to me how to approach this problem?
Thank you.
10
votes
3answers
123 views
Product of two algebraic varieties is affine… are the two varieties affine?
Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then?
If this is not true, could you give a counterexample?
0
votes
2answers
37 views
General formula for $\prod (x+a_i)$
What could be a general formula for this in terms of $x$ and $a_1,\; a_2\; \ldots\; a_{n-1}, \;a_n$?
$$\prod_{i=0}^n(x+a_i)$$
I've tried solving it, but I'm lost at the ...
3
votes
2answers
36 views
Summation and Product Bounds
If I have a sum or product whose upper index is less than its start index, how is this interpreted? For example:
$$\sum_{k=2}^0a_k,\qquad \prod_{k=3}^1b_k$$
I want to say that they are equal to the ...
2
votes
0answers
50 views
“Product” bundle notation.
Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively.
Then there is an induced ...
0
votes
1answer
31 views
3 equations with 9 unknown variables with scalar product
Excuse my bad english pls. I can't find a proper solution to my problem because i don't know the exact mathematical terms in english.
My problem is how to get the 3 elements of each of 3 vectors ...
1
vote
1answer
55 views
Is the product of two measurable subsets of $R^d$ measurable in $R^{2d}$?
Suppose that $E_1,E_2$ are two measurable (Lebesgue) subsets of $R^d$. Define $E=E_1\times E_2=\left\{(x,y)|x\in E_1, y\in E_2\right\}$. Can we say that $E$ is a Lebesgue measurable subset of ...
2
votes
1answer
53 views
Proving a poset is atomic
A poset $(X,\le) $ is atomic if it has both a smallest and largest element, it is graded ,and every element $x$ of $X$ is the join $x_1\vee \dots\vee x_n$ of some elements of $X$ (also written as ...
0
votes
1answer
24 views
Showing an indentity with a cyclic sum
let $z_1,z_2,..,z_n$ be non equal complex numbers
for any $n\geqslant2$, for any $k\in \mathbb{N}$
$$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j \ne i}}^{ n }{ ...
5
votes
0answers
40 views
Is there another, better way to write the following product?
I have the following expression
$$ \prod_{k=0}^n k + \alpha(-1)^{k+1} $$
which is, for example, $(0-\alpha)(1+\alpha)(2-\alpha)$ for $n = 2$. Is there a way to write this using something like a ...
2
votes
1answer
35 views
How to compute a product of logarithms?
I've been reading through Stewart's Calculus textbook, and came across the following problem fairly early on -
What is $$\prod_{i = 2}^{31} \log_i (i + 1)\;?$$
I did some searching, and found ...
7
votes
2answers
121 views
Is $ \prod\limits_{k=0}^\infty \left(1 + \frac{1}{k!}\right) = \mathrm e^2 $?
I was playing around and I came up with this product, which I believe to be equal to $\mathrm e^2$.
$$ \prod_{k=0}^\infty \left(1 + \frac{1}{k!}\right) \stackrel{?}{=} \mathrm e^2 $$
After ...
0
votes
1answer
39 views
Use Proof By Induction to find the product of consecutive odd integers up to $2n-1$
I'm a bit stuck on this inductive proof. I have to find what this is equal to. Product of $1 \times 3 \times 5 \times \ldots \times (2n-1)$ Starting with $i= 1$. What would be a good starting point?
3
votes
1answer
38 views
does invertibility of product imply invertibility of each term of product?
Suppose $\mathcal{H}$ is a Hilbert space and the product $T_1T_2 \in B(\mathcal{H})$ is invertible. Does this imply that both $T_1, T_2$ are invertible ? I am unable to prove this since, unlike the ...
-2
votes
4answers
96 views
Derivative of product notation?
Presume $f(x,y)$ is a continuous function. How would I take the derivative of $$\prod_{x=1}^N f(x,y)$$?
Edit: derivative with respect to $x$, that is.
2
votes
0answers
53 views
Pi identity with sum and product
Please prove this identity
$$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
0
votes
1answer
47 views
Countable product of finite sets with a new metric, compact?
Suppose we have a finite set $E$. Is it true that $E^n$ is compact?
The metric on $E^n$ is :
$$d(\omega,\omega\prime)=\begin{cases}2^{-\inf \{ n \in \mathbf N:\omega _n \ne \omega'_n\} }&{\omega ...
1
vote
1answer
91 views
Prove that if $η$ is exact, then $η∧β$ is also exact.
Prove that if $η$ is exact, then $η∧β$ is also exact.
Please give a clear way to solve?
1
vote
2answers
96 views
Maximize the product of linear functions
Suppose $f(x,y) = \prod_{i=1}^n (a_ix+b_iy)$
where $n$ is a constant larger than 500, and $a_i>0$, $b_i>0$ are known coefficient. There is only one global maximum.
What's the most efficient ...
14
votes
3answers
240 views
Finding $ \prod_{n=1}^{999}\sin\frac{n \pi}{1999}$
I would appreciate if somebody could help me with the following problem. How can we find the product
$$ \prod_{n=1}^{999}\sin\frac{n \pi}{1999}$$
2
votes
1answer
31 views
proving than an infinite product defines an entire function
Consider the infinite product
$$F(z)=\displaystyle\prod_{n=1}^{+\infty}(1-e^{-2\pi n}e^{2\pi iz})$$
How can i prove that $F$ is entire? Can i write $F$ as a Weierstrass product $\prod ...
1
vote
0answers
52 views
Bounding the product of a sequence
I am trying to find an upper bound for the following sequence:
$$(1-p_1)(1-(p_1+p_2))\cdots(1-(p_1+\cdots+p_n))$$
with $n$ groups to multiply. I have written it like this:
$$\prod_{i=1}^n \left({1 ...
3
votes
1answer
81 views
Deducing that “the probability of the intersection is (or is not) the product of the probabilities” from knowledge about other intersections
Let $A_1, A_2, \ldots, A_n$ be a collection of events in a probability space. There are $2^n - n - 1$ subsets S of $\{1, 2, \ldots, n\}$ for which we may or may not have $P(\bigcap_{j \in S}A_j) = ...
13
votes
1answer
376 views
How does one calculate the product of $\tan 1^{\circ} … \tan 45^{\circ}?$
I have seen a question asked on yahoo asking to find the value of
$\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \dots \cdot \tan 45^{\circ}$ (in degrees)
I have seen various results concerning ...
9
votes
3answers
371 views
Prove this product
How to prove this product?
$$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
1
vote
0answers
44 views
Analytic Integration of product of exponential families
I'm happy to join your community and I hope you can help me solve this seemingly straightforward dilemma I am facing. For my thesis, I am trying to solve analytically a product of two distributions ...
8
votes
2answers
176 views
How to find finite trigonometric products
I wonder how to prove ?
$$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$
give me a tip
0
votes
1answer
37 views
Weak direct product
I am just reading the book "Algebra" by Hungerford and on one page it says that
if $G_i$ is a family of groups $\forall i\in I$ then $\prod_{i\in I}^{w}G_i\unlhd\prod_{i\in I}G_i$
where ...
12
votes
5answers
378 views
Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$
Evaluate
$$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$
I can't see anything in this limit , so help me please.
4
votes
2answers
75 views
product of two distinct squares
Is there any shorter and efficient way to find
if a number can be formed by the product of two distinct square numbers
for example
36=4*9
144=16*9
help me with an algorithm or the logic
1
vote
1answer
190 views
Convergence of infinite product
This could be something which is already somewhere in the website, but I am unable to locate any.
Prove $$\prod_{n=1}^{\infty} (1-z^n)$$ converges absolutely and uniformly on each compact subset of ...
2
votes
4answers
127 views
The product $\prod_{m=1}^{11} (x^m - m)$
What would be the co-efficient of $x^{60}$ in the expansion of $\space$ $\prod_{m=1}^{11} (x^m - m)$ ?
1
vote
1answer
40 views
Asymptotics of a Product of Rational Expressions
The following is taken from page 8 of Alon and Spencer's The Probabilistic Method.
$$ \prod_{i = 0}^{n-1} \frac{v - 2i}{v-i} \sim e^{-n^2/2v} $$ as long
as $v \gg n^{3/2}$, estimating
...
4
votes
1answer
108 views
Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$
Difficult question from some test somewhere (I forget).
$$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$
$x$ is, of course, an integer.
1
vote
3answers
126 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 X1 and X2 below) of objects ...
1
vote
1answer
59 views
Infinite Product is converges
I am adding this problem since it is interesting and valuable to be verified here:
Prove that the infinite product $\prod_{k=1}^{\infty}(1+u_k)$, wherein $u_k>0$, converges if ...
2
votes
1answer
71 views
Limit of an n-ary product
Since a definite integral is defined as
$$\lim_{n\to\infty} \sum_{i=0}^n f(x_i^*)\,\Delta x = \int_a^b f(x)\,dx$$
and the integral is much easier to calcluate than a sum, if we change the sum to a ...
4
votes
1answer
114 views
Operators - sums, products, exponents, etc.
$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$.
$(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$.
Is there an operator, such that if ...
2
votes
1answer
103 views
Product of 3 Matrices
Is this identity true for all possible values of $A,B$ and $C$?
$$A^TBC = C^TBA$$
where either
$A,B,C$ are square matrices of same size
$A$ and $C$ are vectors of size $n$ and $B$ is square matrix ...
12
votes
2answers
232 views
proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$
i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows:
$$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
1
vote
1answer
131 views
Homeomorphism of product of topological spaces
I am stuck on a problem about homeomorphic topological spaces and can't go on...
So the problem is:
If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...
0
votes
0answers
61 views
Simplifying very large Euler Product
I am trying to evaluate a very large euler product in terms of the zeta function can someone help me pull out factors and rewrite them in terms of the zeta function, I dont think the whole thing can ...
4
votes
2answers
146 views
Are Euclid numbers squarefree?
Are Euclid numbers squarefree ?
An Euclid number is by definition a Primorial number + 1.
See http://mathworld.wolfram.com/Primorial.html.
In notation the $n$ th Euclid number is written as $E_n = ...
0
votes
2answers
193 views
the limit of infinite product $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots $
I wonder if the function $(1+y)(1+y^2)(1+y^3)(1+y^4)\cdots, 0< y<1$, converges to some well-known function.
If we let $ (1+y)(1+y^2)(1+y^3)(1+y^4)\cdots = \prod_{i=1}^\infty (1+y^i) =
...
2
votes
0answers
123 views
Product of sines
I am looking to evaluate
$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n}$$
without using complex numbers. I can show the result if $n$ is a power of $2$, but if $n$ is anything else I reach a point where I ...
