For questions on infinite products: convergence, computation, etc...

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4
votes
3answers
618 views

Why don't infinite sums make any sense?

Using the infinite sum series, an infinite sum of (1/5)to the nth power, where n goes from zero to infinity, the general summation equation tells us that the answer is 5/4. However, how is this ...
0
votes
1answer
31 views

Is there any identity for this series?

While solving inequality and finite series problem I often come across this series- $$(n+1)(n+2)(n+3)...(n+n)$$. Is there a general solution to this form of a series? Thanks for any help!!
0
votes
0answers
33 views

When is the limit of an infinite product equal to the infinite product of the limit?

For a finite case we have $\lim\limits_{n\rightarrow\infty}f(n)\cdot g(n) =\lim\limits_{n\rightarrow\infty}f(n)\cdot\lim\limits_{n\rightarrow\infty}g(n)$ however when is it possible to interchange the ...
3
votes
1answer
73 views

Is there a name for an infinite product series?

I already know about the Harmonic series: $$\sum_{n = 1}^{\infty} \frac 1n = 1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \cdot \cdot \cdot$$ But is there a name for this infinite ...
0
votes
1answer
31 views

Finding an Entire function with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$

I am really stuck on a homework problem, which boils down to the following: We need to exhibit an entire function $f$ with $f(n \text{ln}(n)) = 0$ for $n \in \mathbb{N}$. The only sorts of functions ...
0
votes
1answer
28 views

Show $\prod_{n=1}^\infty 1 + \frac{-\left( 1 + z \right)}{n^2 + \left( 1 - n^2 \right) z}$ has no analytic extension past the unit disk

I'm studying for a qualifying exam (tomorrow) and I was hoping someone could show me how to finish solving this problem. Let \begin{align} a_n = 1 - \frac{1}{n^2}, && f(z) = ...
0
votes
1answer
32 views

Alternative proof of $\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it}$

Let $t\in \mathbb{R}$. I want an alternative proof of the following identity $$\prod_{n=1}^{+\infty}\frac{e^{it/2^n}+1}{2}=\frac{e^{it}-1}{it} \quad(\star)$$ I've came up with this identity observing ...
2
votes
3answers
114 views

Infinite exponentials

We can read a lot of about convergence of series or Infinite products. E.g. for series. Following series $$\sum_{i=1}^\infty a_i$$ is convergent when $$\lim_{n\rightarrow\infty}a_n=0$$ and ...
6
votes
2answers
52 views

When An Infinite Product Topology Is Hausdorff?

Is the following true: Suppose $(X_1, \tau_1)$ is Hausdorff while $(X_2, \tau_2)\space ...$ are not. $x, y \in X_1 \space\space U,V\in\tau_1\space x \in U , y \in V \space\space V \cap U = \emptyset ...
0
votes
2answers
127 views

Show that $f(x^a) = a f(x) $ and $f(x y) = f(x) + f(y)$

Let $f(x) = (x-1) \large\prod_\limits{n=1}^{\infty} \dfrac{2}{x^{2^{-n}} + 1} $ For real $x > 0$ it is easy to show that $f(x^2) = 2 f(x)$. Let $a$ be a real number. Question 1 Show that ...
-1
votes
0answers
46 views

Need help computing $\prod_{n=2}^\infty(1-n^{-2})$ [duplicate]

I need some help finding product. I am new to this, so I need some help. I am trying to compute $\prod_{n=2}^\infty(1-n^{-2})$. Please help me with this.
2
votes
4answers
60 views

Paradox: Summation of natural logarithms

Consider the expression : $$\sum_{i=1}^{\infty}\ln(i+2)-\ln(i+4)$$ If one evaluates it out, one gets $$\ln(\frac{3\times4\times5\times6\times...}{5\times6\times7\times8\times...})=\ln(12)$$ That ...
1
vote
1answer
50 views

Infinite product $\prod_k (1-z/a_k)(1-z/b_k)^{-1}$ converges uniformly on compacts of$\{\mathbb Im z>0\}$ and is Herglotz

Hi everyone I find the following exercise of infinite product functions.I'd stuck in this exercise. Let $\{a_k\}$ and $\{b_k\}$ sequences of positive real numbers such that $b_k<a_k<b_{k+1}$ ...
8
votes
6answers
149 views

Does the multiplication of countably infinite many numbers between $0$ and $1$ equal $0$?

Suppose every term of a countably infinite sequence $x_1,x_2,\dots$ is between $0$ and $1$, i.e. $0<x_i<1$ for every $i$. Does $\mathop {\lim }\limits_{n \to \infty } \prod\limits_{i = 1}^n ...
2
votes
1answer
23 views

infinite product of polynomials

Let $p_k$ be a complex polynomial with $p_k(0)=1$, with no zero in $\{|z| \le k^3\}$ and $deg(p_k) \le k$. Show that $\prod_{i=1}^\infty p_i(z)$ converges. One can observe that $p_k(z)=1+h_k(z)$ but ...
1
vote
0answers
26 views

An explicit formula for some hadamard product

Is there an explicit formula for $$\prod_{k=1}^{\infty} \Big(1- \dfrac{z}{\pi^{3/2}\;k^{3/2}}\Big) $$ ?
1
vote
0answers
32 views

Absolute convergence of $\prod(1-z^n)$

I'm trying to find the radius of convergence of $\prod_n^\infty(1-z^n)$. Using the result that "$\sum f(x)$ converges abs. $\to$ $\prod (1+f(x))$ converges abs." with $f(x)=-z^n$. And the ...
2
votes
3answers
74 views

How do I evaluate the sum $\sum_{k=1}^\infty\left(\ln\big(1+\frac{1}{k+a}\right)-\ln\left(1+\frac{1}{k+b}\big)\right)$ [closed]

How do I evaluate the sum $$\sum_{k=1}^\infty\left(\ln\Big(1+\frac{1}{k+a} \Big)-\ln\Big(1+\frac{1}{k+b}\Big)\right)$$ where $0 <a<b<1$? Hints will be appreciated Thanks
5
votes
2answers
64 views

Issues solving equations involving $x^{x^x…}$?

I stumbled across this problem: $x^{x^{x^{...}}}=2$ Obviously, I used the substitution trick and I got $x^2=2$ and thus, $x=\pm\sqrt{2}$. I have tested that this works. However, I tried to ...
4
votes
2answers
130 views

Infinite product involving primes

I just had my first analysis course as an undergraduate, and I'm trying to learn more about analytic number theory. Right now I'm looking at prime numbers in particular--I'm studying (mostly just ...
5
votes
1answer
104 views

Prime-twins and infinite products

For $n\geq 1$ let the nth twin prime pair $$(p_n,p_n+2).$$ This sequence start as $(3,5),(5,7)$, the next $(11,13)\ldots$. I have two short questions about twin primes and infinite product defined ...
12
votes
1answer
165 views

Asymptotics for a series of products

I am trying to solve the following problem: Define the following functions for $x>0$: $$f_n(x):=\prod_{k=0}^{n}\frac{1}{x+k}$$ Show that the function ...
2
votes
0answers
38 views

How do I show that $\prod_{n=2}^\infty1-n^{-z})$ converges?

What are $z$'s such that $\prod_{n=2}^\infty (1-n^{-z})$ is convergent? Let $\mathscr{R}$ be the region where the product converges. Assume that the product is convergent at $z$. Then, ...
1
vote
1answer
51 views

How can I calculate the partial product formula of $\prod_{k=2}^n\left(\frac{k^3-1}{k^3+1}\right)$

Show that $\prod_{k=2}^{\infty} \left(\frac{k^3-1}{k^3+1}\right)=\frac{2}{3}$ Ok so in Remmert's Classical Topics in Complex Function Theory In the introduction to infinite products, there is ...
0
votes
0answers
31 views

Reduce to Wallis formula

How do we get Wallis Formula $$\frac{\pi}{2}=\lim_{l\to\infty} \prod_{j=1}^{l+1}\frac{(2j)(2j)}{(2j-1)(2j-1)} $$ from $$\lim_{n\to\infty}\frac{(n+1)^2}{n+\frac{3}{2}} ...
0
votes
0answers
21 views

An exercise concerning infinite products

I am stuck with the following exercise: let $\alpha$ be a complex number which is not integer. Prove $$\frac{\sin\pi(z+\alpha)}{\sin\pi\alpha}=e^{\pi z\cot\pi\alpha}\prod_{n=-\infty}^\infty ...
7
votes
2answers
127 views

Infinite product equality $\prod_{n=1}^{\infty} \left(1-x^n+x^{2n}\right) = \prod_{n=1}^{\infty} \frac1{1+x^{2n-1}+x^{4n-2}}$

Prove the following equation ($|x|<1$) $$\prod_{n=1}^{\infty} \left(1-x^n+x^{2n}\right) = \prod_{n=1}^{\infty} \frac1{1+x^{2n-1}+x^{4n-2}}$$ I made this question and I have the following answer ...
1
vote
1answer
21 views

Difference of Divergent Products Converges?

I was thinking about series and was wondering how infinite products are dealt with. For example, consider this difference of two divergent products: $$\prod_{n=2}^\infty \ln(n) - \prod_{n=2}^\infty ...
4
votes
0answers
108 views

Evaluate $\lim\limits_{n\to \infty} {\frac{(2n-1)!!}{(2n)!!}}$ [duplicate]

I have tried using Stolz–Cesàro's formula and subtract the next term in the series but that gave me $$\lim_{n\to \infty} {\frac{2n}{2n+1}}$$ which is obviously 1 and not right. I do realise that the ...
2
votes
1answer
51 views

Product of Uncountably Infinite Number of 1s

Just like the title says: What is the product of an uncountable number of 1s? Intuitively the answer is 1, but how does one go about defining such a product in general?
1
vote
2answers
59 views

Infinite product $\prod\limits_{k=0}^\infty\sum\limits_{n=0}^9z^{10^kn} $ leading to $1/(1-z)$

Please give me a hint (i am studying Complex Variables for Engineering) on how to prove that ...
-4
votes
1answer
36 views

Calculate the limits of sum and products [closed]

Calculate the limits of: $$ u_n= \prod_{i=1}^n (1+\frac{i}{n}) $$ and $$ u_n =\sum_{k=0}^n (C^k_n)^{-1} $$ Thank you a lot. This I my first time posting questions. Tell me if there is something ...
6
votes
1answer
118 views

Pre measure for an infinite product of measure spaces

Let $\{(\Omega_k, \Sigma_k, P_k)\}_{k\geq 1}$ be a sequence of probability spaces. I am trying to prove the statement below in order to use it and get a pre measure and then use the Hahn kolomogrov ...
1
vote
1answer
32 views

Show that a martingale is not $L^1$ convergent

Consider the symmetric random walk $S_n$ on $\mathbb{Z}$. The process $Z_n=\exp(uS_n-n \ \log(\cosh(u)))$ for $u\in \mathbb{R}$ is a positive martingale with $E(Z_n)=1$ for all $n\geq 1$. $Z_n$ is ...
0
votes
0answers
46 views

Calculation of infinite product

My question is to prove the identity: $$ \prod_{n=1}^{\infty}\left(\frac{\cos t-1}{n}+1\right)=\exp\left(-\int_0^1x^{-1}(1-\cos xt)dx\right) $$ which arises as a product of characteristic functions of ...
6
votes
2answers
161 views

Closed form of the integral ${\large\int}_0^\infty e^{-x}\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)dx$

While doing some numerical experiments, I discovered a curious integral that appears to have a simple closed form: $${\large\int}_0^\infty ...
7
votes
1answer
82 views

How to prove this infinite product identity?

How can I prove the following identity? $$\large\prod_{k=1}^\infty\frac1{1-2^{1-2k}}=\sum_{m=0}^\infty\left(2^{-\frac{m^2+m}{2}}\prod_{n=1}^\infty\frac{1-2^{-m-n}}{1-2^{-n}}\right)$$ Numerically both ...
2
votes
0answers
33 views

A variation of Viète's formula — an infinite product of nested radicals [duplicate]

Viète's formula expresses an infinite product of nested radicals in terms of $\pi$. Let $$a_1=\sqrt2,\quad a_n=\sqrt{2+a_{n-1}}.$$ Note that $\lim\limits_{n\to\infty}a_n=2.$ Then ...
0
votes
1answer
175 views

Joint distribution of an infinite collection of random variables?

Let's say we have a countable collection of random variables $X_1, X_2, ...$, in $(\Omega, \mathscr{F}, \mathbb{P})$ Can we define a joint distribution function for all of them ie $$F_{X_1,X_2, ...
1
vote
3answers
61 views

Prove that $\prod\limits_{k=0}^n\frac{3k+1}{3k+2}\to0$ by elementary means

I'm trying to find limit of sequence $s_n$, but with only elementary operations (eg. without integrals), like basic multiplication or logarithms. $$x_n = \frac{3n+1}{3n+2} \wedge s_n = ...
9
votes
1answer
274 views

Is $\prod_{n=1}^\infty P_{2n-1}$ regularizable?

Assume that $P_n$ denotes the $n$'th prime for this entire question. Inspriation: I was dumbfounded by the fact that: $$\hat\prod_\limits{n=1}^\infty P_{n}=4\pi^2$$ After further investigation, I ...
4
votes
0answers
68 views

Why is the infinite product of this quotient of $\sin$'s equal to $\left(\frac{3}{\pi}\right)^{2}$[SOLVED]

I was intrigued by this answer the other day, but perhaps lack a little bit of the necessary background to understand a certain step. Namely, the fact that $$\prod_{n=1}^{+\infty} ...
2
votes
1answer
76 views

What does this product converges to?

Let $p\in[0,1]$. I'm interested in computing $$\lim_{n\to\infty}\prod_{i=1}^n(1-p^i)$$ Any thoughts? EDIT: As Kibble mentioned, this is the Euler function. Also from Kibble: a simple upper ...
3
votes
3answers
160 views

Limit of a sum of infinite series

How do I find the following? $$\lim_{n\to\infty} \frac{\left(\sum_{r=1}^n\sqrt{r}\right)\left(\sum_{r=1}^n\frac1{\sqrt{r}}\right)}{\sum_{r=1}^n r}$$ The lower sum is easy to find. However, I don't ...
8
votes
1answer
460 views

Evaluate the infinite product

Let $A_0=\dfrac{3}{4}$, and $A_{n+1}=\dfrac{1+\sqrt{A_n}}{2}$ for all $n\geq0$. How to find the value of $\displaystyle\prod_{n=1}^\infty A_n$ ? I don't have any idea. Thank you.
2
votes
1answer
37 views

First non-zero digit

In terms of n, which is the first non-zero digit of $\prod\limits_{i=1}^{n/2} (i)(n-i+1)$ for even n $\geq$ 6? Thank for any advice.
7
votes
2answers
299 views

Find the value of $(1-z)\left(1+\frac{z}{2}\right)\left(1-\frac{z}{3}\right)\left(1+\frac{z}{4}\right)\cdots$.

Show that: $$(1-z)\left(1+\frac{z}{2}\right)\left(1-\frac{z}{3}\right)\left(1+\frac{z}{4}\right)\cdots = ...
0
votes
0answers
30 views

Stuck in proof of Lemma 5.8 in Conway's Functions of one complex variable I

5.8 Lemma Let (X,d) be a compact metric space and let $\{g_n\}$ be a sequence of continuous functions from X into $\mathbb{C}$ such that $ \sum g_n(x)$ converges absolutely and uniformly for x in X. ...
3
votes
3answers
54 views

How do I compute $\prod_{n=1}^\infty \mathrm{e}^{{\mathrm{i}\pi}/{2^n}}$?

I'm struggling with how to compute the following product: $$\prod_{n=1}^\infty \mathrm{e}^{{\mathrm{i}\pi}/{2^n}} $$ Wolfram Alpha tells me it's $-1$, and I can confirm that it converges since ...
0
votes
0answers
34 views

If $\sum\limits_n|1-a_n|$ converges then $\prod\limits_n a_n$ converges

Let $(a_n)_n$ be a positive real sequence. I want to show that if $\sum\limits_n|1-a_n|<+\infty$ then $\prod\limits_n a_n$ converges and it's positive. It is sufficient to show that $\sum\limits_n ...