For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.
21
votes
2answers
477 views
Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$
I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating ...
20
votes
4answers
820 views
What is to geometric mean as integration is to arithmetic mean?
The arithmetic mean of $y_i ... y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$
For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using ...
16
votes
3answers
661 views
Result of the product $0.9 \times 0.99 \times 0.999 \times …$
My question has two parts:
How can I nicely define the infinite sequence $0.9,\ 0.99,\ 0.999,\ ...$? One option would be this recursive definition below; is there a nicer way of doing this? Maybe ...
15
votes
4answers
1k views
Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$
While trying some problems along with my friends we had difficulty in this question.
True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
14
votes
3answers
721 views
A “fast” way for computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $?
Which is the fastest paper-pencil approach to compute the product $$ \prod
\limits_{i=1}^{45}(1+\tan i^\circ) $$
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
$$ \prod_{n=1}^{999}\sin\frac{n \pi}{1999}$$
13
votes
1answer
273 views
A question about $\prod_{x\in \mathbb{R}^{*}}{x}$
When I was an elementary school student, I encountered some puzzles like "What's product of numbers of hair of each people in the world?", which answer was zero, because there's people with no ...
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 ...
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.
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:
$$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
11
votes
2answers
495 views
Infinite products - reference needed!
I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to ...
10
votes
3answers
123 views
Product of two algebraic varieties is affine… are the two varieties affine?
Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then?
If this is not true, could you give a counterexample?
10
votes
0answers
172 views
When is an infinite product of natural numbers regularizable?
I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like
$$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$
and
$$\infty ...
9
votes
4answers
619 views
Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$
Recently, I ran across a product that seems interesting.
Does anyone know how to get to the closed form:
$$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$
I ...
9
votes
3answers
371 views
Prove this product
How to prove this product?
$$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
8
votes
7answers
557 views
Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?
I am trying to construct a sequence $\{x_{n}\} \in (0,1)$ such that such that the product of all its terms is $\frac{1}{2}$.
Please can I have any clue to solve my problem?
Thanks.
8
votes
2answers
175 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
8
votes
0answers
157 views
How to find the the product $\left(1 - \frac{1}{a}\right)\left(1-\frac{1}{a^{2}}\right)\left(1-\frac{1}{a^{3}}\right)\ldots$ [duplicate]
Possible Duplicate:
Result of the product $0.9 \times 0.99 \times 0.999 \times \dots$
How to find the product $$\left(1 - ...
7
votes
1answer
249 views
Evaluation of a product of sines [duplicate]
Possible Duplicate:
Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
I am looking for a closed form for this product of sines:
\begin{equation}
\sin ...
7
votes
2answers
310 views
Proving an infinite product formula
I have found this formula and I am trying to prove it , but I have not any idea how to deal with it:
$$e^{ax}-e^{bx} = ...
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 ...
7
votes
1answer
123 views
Zeros in the complex plane and convergence
I'm doing some number theory which requires some work in $\mathbb{C}$, but unfortunately my complex analysis is a little rusty.
A text I am reading states the following:
...and given that ...
6
votes
5answers
276 views
Definition of the Infinite Cartesian Product
(1) If $X$ and $Y$ are two sets, we define the Cartesian product $X \times Y$ as the set of ordered pairs $(x,y)$, such that $x \in X$ and $y \in Y$.
(2) On the other hand [Folland, Real Analysis, ...
6
votes
1answer
269 views
Generalization of the series for $\frac{\pi^2}{6}$? Is there a more elementary proof?
In the same vein as:
$ \frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots $
Starting with:
$ \displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = ...
6
votes
3answers
105 views
closed-form expressions for product of 3n+k where k = 1 or 2
There are some easy products that can be written in closed form in terms of factorials:
$ 2 \times 4 \times 6 \times ... 2n = n! \times 2^n$
$ 1 \times 3 \times 5 \times ... (2n-1) = {{(2n)!} ...
6
votes
1answer
177 views
Which is the Abel's theorem invoked in the context of convergence of this infinite product?
Motivation: As I wrote in this answer the following product is evaluated in Proposition 5 of Jean-Paul Allouche and Jeffrey Shallit's paper The
ubiquitous Prouhet-Thue-Morse sequence
...
5
votes
2answers
194 views
Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$
$f(x)=\sum\limits_{k = 2 }^ \infty e^{-kx} \ln(k) $
$\int\limits_0^{\infty}\int\limits_x^{\infty}\, f(\gamma)\, d\gamma dx=\sum\limits_{k = 2 }^ \infty \frac{1}{k^2} \ln(k) $
...
5
votes
2answers
302 views
How was Euler able to create an infinite product for sinc by using its roots?
In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that
$$\begin{align*}
\frac{\sin(x)}{x} &=
\left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
5
votes
1answer
247 views
Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?
If $a_1\ge a_2 \ge a_3 \ldots $ and if $b_1,b_2,b_3\ldots$ is any rearrangement of the sequence $a_1,a_2,a_3\ldots$ then for each $N=1,2,3\ldots$ one has
$$\sum^N_{n=1}\left(\prod_{i=1}^n b_i ...
5
votes
1answer
186 views
Uncountable product in the category of metric spaces.
I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces.
Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...
5
votes
1answer
116 views
Continuous maps from products of topological spaces
Let $X,Y,Z$ be topological spaces. It is well-known that if $F:X\times Y\to Z$ is a continuous map, we can define a map $$\overline{F}:X\to C(Y,Z) \\\overline{F}(x)(y)=F(x,y)$$ where $C(Y,Z)$ is the ...
5
votes
1answer
99 views
Modulus of infinite product of complex functions
We know that for complex (entire) functions $f,g$ we have $|f(z)g(z)|=|f(z)||g(z)|$, where $|.|$ means complex modulus.
What about if we have an infinite product? Is it true that
$$\bigg| ...
5
votes
0answers
40 views
Is there another, better way to write the following product?
I have the following expression
$$ \prod_{k=0}^n k + \alpha(-1)^{k+1} $$
which is, for example, $(0-\alpha)(1+\alpha)(2-\alpha)$ for $n = 2$. Is there a way to write this using something like a ...
4
votes
2answers
465 views
How to evaluate $\lim\limits_{n\to+\infty} \prod\limits_{k=1}^n (1+k/n^2)$?
I've got a limit which puzzle me several days. The question is
$$ \lim_{n\to+\infty} \prod_{k=1}^n\left(1+\frac{k}{n^2}\right).$$
Can you help me? Thank you in advance
4
votes
4answers
86 views
Big Greeks and commutation
Does a sum or product symbol, $\Sigma$ or $\Pi$, imply an ordering?
Clearly if $\mathbf{x}_i$ is a matrix then:
$$\prod_{i=0}^{n} \mathbf{x}_i$$
depends on the order of the multiplication. But, ...
4
votes
1answer
286 views
When is $\displaystyle \prod_i \prod_j a_{i} a_{j} = \Bigl(\prod_i a_i\Bigr)^2$
In statistical mechanics, I used to use the procedure that if $a_{ij}=a_i a_j$ $$\prod_i\; \prod_j a_{i}a_{j} = \biggl(\prod_i a_i\biggr)\vphantom{\Bigr)}^2$$
However, today I noticed, $$\prod_i\; ...
4
votes
2answers
236 views
Infinite product
How do I solve the infinite product of $$\prod_{n=2}^\infty\frac{n^3-1}{n^3+1}?$$
I know that I have to factorise to $$\frac{(n-1)(n^2+n+1)}{(n+1)(n^2-n+1)},$$
but how do I do the partial product?
...
4
votes
2answers
73 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
4
votes
1answer
162 views
Is there a “continuous product”?
Is there a "continuous product" which is the limit of the discrete product $\Pi$, just like the integral $\int$ is the limit of summation $\sum$.
Thanks!
4
votes
1answer
240 views
What is $\prod_{k=1}^n (1-x^k)$?
I'd like to know what
$$\prod_{k=1}^n (1-x^k)$$
evaluates to (assuming there is a simple closed form) and what it "is" in the context of commutative algebra (of which I knew little and recall ...
4
votes
2answers
152 views
When equal products imply equal factors?
Under which additional conditions $a\times b = c\times d \Rightarrow a=c\wedge b=d$ (where $\times$ is a categorical product)?
For example, in the case of Cartesian product, for this is enough when ...
4
votes
2answers
172 views
Closed form expression for a product.
A simple method for evaluating a product is term cancellation. For example, the product
$$\begin{align*}
\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)&=\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)\\
...
4
votes
1answer
114 views
Operators - sums, products, exponents, etc.
$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$.
$(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$.
Is there an operator, such that if ...
4
votes
1answer
108 views
Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$
Difficult question from some test somewhere (I forget).
$$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$
$x$ is, of course, an integer.
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 = ...
4
votes
1answer
288 views
Dyson series and T product
One of the most important tool in quantum mechanics is the Dyson series because it is the basis of the perturbative theory. There is a step in the derivation that I can't understand.
$\{H(t_i)\}$ are ...
4
votes
1answer
77 views
Products in the category of sets and (left-)total relations
By a total (or left-total) relation I mean a binary relation $R \subseteq X \times Y$ where there is, for each $x \in X$, at least one $y \in Y$ with $(x,y) \in R$. Equivalently stated, I mean ...
4
votes
0answers
82 views
Product-Decomposition of distributive lattices
Every nontrivial (bounded) distributive lattice arises as a direct power of a certain number of nontrivial product-irreducible (bounded) distributive lattices. My question is how this number can be ...
3
votes
5answers
593 views
The limit of infinite product
Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
3
votes
2answers
78 views
Calculate $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
Please help me solving $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
