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

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3
votes
1answer
168 views

Partition Proof

Let $\lambda$ be a partition of $N$ of rank $r$. How can I show that: $$\sum_wx^\lambda(w)=f^\lambda(-1)^{t(\lambda)}\prod^r_{i=1}(\lambda_i-1)!(\lambda'_i-1)!$$ where $w$ ranges over all ...
0
votes
1answer
65 views

Product rule question about Alphabet

I am trying to understand the product rule and I have a simple example it says, ...
2
votes
1answer
85 views

How to calculate a bound for this product?

Consider the following product: $$ \prod_{i=1..n} {\left(1 - {1 \over 2^i}\right)} $$ A numeric calculation, up to $n=20$, gives $0.288788370496567$. But how can I calculate its limit when $n$ goes ...
23
votes
4answers
726 views

Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
1
vote
1answer
8k views

Magnitude of a Matrix?

Consider a vector V. The magnitude of this vector (if it describes a position in euclidean space) = distance from the origin is simply: $(V^TV)^{1/2} $ aka the square root of the dot product... ...
0
votes
1answer
118 views

Expression for sum of $k$-products of $n$ variables

Given $n$ variables there are $n \choose k$ different terms that are the product of $k$ different variables. For example, in the case that $n = 3$, the $k$-products of the variables $x_1, x_2, x_3$, ...
3
votes
1answer
418 views

A product identity involving the gamma function

I have reduced this problem (thanks @Mhenni) to the following (which needs to be proved): ...
1
vote
1answer
94 views

faithful irreducible representation of $A_{4} \times Q_{8}$

Construct a faithful irreducible representation of the group $A_4 \times Q_8$ $A_{4}$ is the alternating group $Q_{8}$ is the quaternions
4
votes
1answer
98 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, \;\;\; ...
3
votes
2answers
864 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 = ...
2
votes
1answer
117 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
100 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
77 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
77 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
355 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
71 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 ...
4
votes
2answers
140 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
110 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 ...
1
vote
1answer
473 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
64 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 ...
7
votes
2answers
219 views

Showing an indentity with a cyclic sum

Let $n\geqslant2$, and $k\in \mathbb{N}$ Let $z_1,z_2,..,z_n$ be distinct complex numbers Prove that $$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j ...
6
votes
1answer
153 views

Is there another way to write the product $\prod_{k=0}^n\left(k+\alpha\left(-1\right)^{k+1}\right)$?

I have the following expression $$ \prod_{k=0}^n \left(k + \alpha(-1)^{k+1}\right), $$ which is, for example, $(0-\alpha)(1+\alpha)(2-\alpha)$ for $n = 2$. Is there a way to write this using ...
2
votes
1answer
67 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 ...
10
votes
2answers
237 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
154 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?
2
votes
1answer
133 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
414 views

Derivative of product notation? [closed]

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.
1
vote
0answers
88 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
158 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
143 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
391 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 ...
20
votes
3answers
500 views

Finding $ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$

I would appreciate if somebody could help me with the following problem. How can we find the product $$ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$$
2
votes
1answer
84 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
84 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
114 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) = ...
12
votes
1answer
565 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
392 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
57 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
250 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
235 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 ...
14
votes
5answers
870 views

Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$ [closed]

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
834 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
2
votes
1answer
315 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
154 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
50 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 ...
5
votes
1answer
138 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.
4
votes
2answers
149 views

Can $\prod\limits_{k=0}^n \left( 2 \cosh(2^kx)-1 \right)$ be simplified?

Do you know if the product $\prod\limits_{k=0}^n \left( 2 \cosh(2^kx)-1 \right)$ can be simplified?
1
vote
3answers
222 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 ...
1
vote
1answer
161 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
136 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 ...