Tagged Questions
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.
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
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 :
$$d(\omega,\omega\prime)=\begin{cases}2^{-\inf \{ n \in \mathbf N:\omega _n \ne \omega'_n\} }&{\omega ...
9
votes
3answers
371 views
Prove this product
How to prove this product?
$$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
8
votes
2answers
175 views
How to find finite trigonometric products
I wonder how to prove ?
$$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$
give me a tip
12
votes
5answers
378 views
Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$
Evaluate
$$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$
I can't see anything in this limit , so help me please.
1
vote
1answer
190 views
Convergence of infinite product
This could be something which is already somewhere in the website, but I am unable to locate any.
Prove $$\prod_{n=1}^{\infty} (1-z^n)$$ converges absolutely and uniformly on each compact subset of ...
4
votes
1answer
108 views
Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$
Difficult question from some test somewhere (I forget).
$$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$
$x$ is, of course, an integer.
1
vote
1answer
59 views
Infinite Product is converges
I am adding this problem since it is interesting and valuable to be verified here:
Prove that the infinite product $\prod_{k=1}^{\infty}(1+u_k)$, wherein $u_k>0$, converges if ...
0
votes
3answers
133 views
Product of sum expression
I am having a little trouble following an example I came across today which says that:
$$2 \sum_{k=1}^{n} \sum_{i=0}^{k-2} 1 = 2 \sum_{k=1}^{n} (k-1) = 2 \sum_{j=0}^{n-1} j$$
I have tried fidgeting ...
3
votes
2answers
78 views
Calculate $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
Please help me solving $\lim_{n\to\infty}[(1+x)(1+x^2)(1+x^4)\cdot\cdot\cdot(1+x^{2n})]$, $|x|<1$
10
votes
0answers
172 views
When is an infinite product of natural numbers regularizable?
I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like
$$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$
and
$$\infty ...
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) $
...
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)!} ...
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 ...
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.
16
votes
3answers
660 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 ...
0
votes
1answer
177 views
How to expand this summation/product $\sum\limits_{n=0}^\infty \prod\limits_{j=0}^{n-1}\frac{\lambda_j}{\mu_{j+1}}$
Let $\mu_n = n\mu$ for $1 \le n \le 3$ and $\mu_n = 3\mu$ for $n \ge 4$. Let $\lambda_n = \lambda$ for all $n \in \mathbb{N}_0$. Define $\rho := \frac{\lambda}{\mu}$.
How would I expand ...
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
$$\begin{align*}
\frac{\sin(x)}{x} &=
\left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
3
votes
2answers
203 views
Given $\sum |a_n|^2$ converges and $a_n \neq -1$, show that $\prod (1+a_n)$ converges to a non-zero limit implies $\sum a_n$ converges.
I have been working on this problem for a while and cannot seem to make any progress without coming up with something wrong or hitting a dead end.
Here is what I have so far:
$ \prod (1+a_n) < ...
4
votes
2answers
171 views
Closed form expression for a product.
A simple method for evaluating a product is term cancellation. For example, the product
$$\begin{align*}
\prod_{k=2}^{n}\left(1-\frac{1}{k}\right)&=\prod_{k=2}^{n}\left(\frac{k-1}{k}\right)\\
...
2
votes
1answer
93 views
Infinite product of recursive sequence
Let $a_{n+1}=\sqrt {(a_n+a_{n-1})/2}$ and $a_0=a_1=2$, how to prove convergence of the product $a_0 a_1 a_2 a_3...a_\infty$, and possibly find its value?
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:
$$e^{ax}-e^{bx} = ...
5
votes
1answer
247 views
Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?
If $a_1\ge a_2 \ge a_3 \ldots $ and if $b_1,b_2,b_3\ldots$ is any rearrangement of the sequence $a_1,a_2,a_3\ldots$ then for each $N=1,2,3\ldots$ one has
$$\sum^N_{n=1}\left(\prod_{i=1}^n b_i ...
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 ...
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
...
2
votes
2answers
518 views
Formula for Geometric Progression
Can someone help me understand the idea behind constructing a formula for the following:
For $n\in\mathbb{N}$, $n\geq 2$, find and prove a formula for:
$$\prod_{i=2}^n \left(1 - ...
3
votes
2answers
55 views
interval for a product to infinity
I was wondering - how would I specify the interval (the amount that n increases each time) between terms? Is that possible? What if I want it to increase by, say, ...
