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Limit of $a_{n+1}=a_{n}\cdot \left( 1+\frac{1}{n}\right)$
Let $a_{n}$ be a recursive sequence such as: $\begin{cases}a_{1}=1\\ a_{n+1}=a_{n}\cdot \left( 1+\dfrac{1}{n}\right) \end{cases}$ I need to show that $\lim _{n\rightarrow \infty }a_{n}=\infty$. I ... sequences-and-series limits infinite-product- 61
Infinite Product of (1-x)^ib, as x approaches 1
Is there a simple form to this product? $\prod_{n=1}^{\infty}(1-\frac{i.b}{n})$ I know for sure that the gamma function relates to it somehow and it could be represented as a binomial series $1-ib+\... infinite-product- 1
Does there exist an example of converging infinite product when all elements are below 1?
I consider the infinite product $$p = \prod_{k=1}^\infty z(k),$$ where $z:\mathbb{N}\to[0,1).\ $ I am wondering if there exists $z$ such that the product converges to a nonzero value? Obviously, we ... calculus sequences-and-series infinite-product- 1,971
Function without holomorphic extension
We know that $\prod_{n=1}^{\infty}\frac{1}{1-z^2/n^2}$ defines an holomorphic function on $U=\{\Im (z) >0\}$, by the equality $\sin (\pi z)=\pi z\prod_{n=1}^{\infty}(1-z^2/n^2)$. Using this ... complex-analysis infinite-product- 35
How to compute $\displaystyle\prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}}$?
In my text book it is stated (without any explanation) that $$ \prod_{i = 1}^{\infty} \frac{x^{i}}{e^{i!}} = e^{e^x - 1} $$ and I can't really think of how one can show this. sequences-and-series infinite-product- 63
Infinite product converging arbitrarily slowly to limit
There evidently exist infinite series that converge to their limits with arbitrary speed. For example, for each $\alpha>0$, the series $$ \sum_{j=1}^n j^{-\alpha-1} \to \sum_{j=1}^\infty j^{-\alpha-... real-analysis calculus limits infinite-product- 1,247
Factorization of modular forms using zeros and poles
In complex analysis, it is possible to decompose any given entire function into a product of linear factors using the zeros, using the Weierstrass factorization theorem. For example, $\sin(z)$ is ... complex-analysis infinite-product modular-forms weierstrass-factorization- 21
Proving that this infinite product is convergent
This question was asked in my assignment in number theory and I could not prove it. Question : Define the multiplicative function w(n) such that $w(p^k) =0$ for $k\geq 2$ and w(p)= { $\frac{p} { f(p)... number-theory analytic-number-theory infinite-product- 1,365
For what infinite series can a closed form be obtained by means of the $\text{Sum} = \text{Product} $ method?
Euler solved the Basel problem by equating the Taylor series and the infinite product representation of $\sin(x)/x$: $$\sum_{n=1}^{\infty}(-1)^{n}\frac{x^{2n}}{(2n+1)!} = \prod_{k=1}^{\infty}\bigg{(} ... real-analysis sequences-and-series reference-request taylor-expansion infinite-product- 5,135
If $\alpha$ is the nth root of unity , then the value of [closed]
Given , $\alpha$ is the nth root of unity . Then , the value of $$(11-\alpha)(11-\alpha^2)(11-\alpha^3)........(11-\alpha^{n-1})$$ is equal to ... I tried to use the property : That sum of nth roots ... sequences-and-series complex-numbers infinite-product- 3
Summing $\sum_{k=1}^{\infty} \frac{(-2)^k + 1}{3^k}$
I’m trying to calculate the sum of this by breaking it down into two geometric series: $$\sum_{k=1}^{\infty} \frac{(-1)^k2^k + 1}{(3^k)} =\sum_{k=1}^\infty\left(-\frac{2}{3}\right)^{k} +\sum_{k=1}^\... infinite-product- 13
How to calculate $\prod _{n=1}^{\infty}\frac{2^n-1}{2^n}$?
Suppose we have a kind of lottery as follow: $1.$ You have a $\frac{1}{2}$ possibility of getting a prize on the first try. $2.$ You have a $\frac{1}{4}$ possibility of getting a prize on the second ... probability limits infinite-product lotteries- 21
Find a formula for calculate $\prod_{n=0}^N(a^n + b)$
I have an expression, in which I am looking for a formula to calculate $$ \prod_{n=0}^N (a^n +b)$$ as a summation. Is there a general name for this type of formula? Or a relevant approach to find ... sequences-and-series products infinite-product- 143
Countable product of complete spaces is complete
Consider Fréchet spaces $\{E_n\}_{n\in\mathbb{N}}$, and let $E = \prod_{n=1}^{\infty} E_n$. Show that $E$ is a Fréchet space. I'm stuck with proving completeness. I have already verified that $E$ is ... infinite-product complete-spaces- 2,246
Factorial limit convergence on e
I'm working with some particular infinite products. Each infinite product applies to a range of real numbers, in much the same way as a Riemann sum applies to a range of real numbers. The specifics of ... infinite-product- 141
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