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

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2
votes
0answers
24 views

Given two entire functions $f_1,f_2$ without common zeros, prove that one can find some entire functions $g_1,g_2$ such that $f_1g_1+f_2g_2=1$ [duplicate]

The question Let $f_1,f_2$ be some entire functions without zeros in common, so for every $z∈ℂ$ we have $|f_1(z)|^2+|f_2(z)|^2≠0$. Prove that there exist two entire functions $g_1,g_2$ such that: $$ ...
3
votes
1answer
60 views

Infinite product include summation

I would like to find an infinite product of$$\prod _{n=2}^{\infty} \left(1+\frac{(-1)^{n-1}}{a_n}\right)$$ where $a_n = \sum_{k=1}^{n-1} \frac{n!(-1)^{k-1}}{k!} $ I tried to compute $a_2 , a_3 ...
-2
votes
1answer
42 views

Infinite Sets Proof [on hold]

I have a few questions regarding this problem below: Prove that if A and B are finite sets, then A ≈ B if and only if |A| = |B|. Would I assume that |A| = |B|? Which would obviously make A ≈ B ...
1
vote
0answers
25 views

Generating functions, Schur's identity

Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...
3
votes
1answer
49 views

Prove that $ζ(4)=π^4/90$ knowing that $\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)$

The question Knowing that: $$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right) \tag{1}$$ obtain the Taylor series expansion of $\frac{\sin(πz)}{πz}$ to deduce: $$ \sum_{1 ≤ n_1 < n_2 ...
0
votes
0answers
44 views

Sequence of non-independent coin tosses

Suppose that a sequence of coin tosses is due to be performed. Let $p_i$ denote the probability that the $i$th coin toss lands on Heads and let $X_i$ denote the corresponding indicator random variable ...
2
votes
1answer
53 views

Knowing the representation of $\frac{π^2}{\sin^2(πz)}$, deduce the Taylor series of $\frac{z^2}{\sin^2{z}}$

The question Consider the representation: $$ \frac{π^2}{\sin^2(πz)} = \sum_{n∈ℤ}\frac{1}{(z+n)^2} \tag{0}$$ valid for all $z ∈ ℂ \setminus ℤ$. Deduce that the Taylor series of $z^2/\sin^2(z)$ for ...
2
votes
1answer
51 views

Hellinger integral properties - proof of equivalence for infinite product measures

I'm trying to prove that: Let $(\mu_k)_{k=1}^{\infty}$ and $(\nu_k)_{k=1}^{\infty}$ be sequences of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Consider the product measures on ...
1
vote
1answer
27 views

Show that $\frac{z}{1-z} = \sum_{j=0}^∞ \frac{2^j}{1 + z^{-2^j}}$ when $z ∈ \mathbb{D}$

The question Knowing that with $z ∈ \mathbb{D}$: $$ \prod_{k=0}^∞(1 + z^{2^k}) = \frac{1}{1-z} $$ prove that with $z ∈ \mathbb{D}$: $$ \sum_{j = 0}^∞ \frac{2^j}{1 + z^{-2^j}} = \frac{z}{1-z} $$ ...
3
votes
2answers
286 views
+50

Show that $\dfrac{\rm{d}^{L-m}}{\rm{d}x^{L-m}}\left(x^2-1\right)^L=\dfrac{(L-m)!}{(L+m)!}(x^2-1)^m\dfrac{\rm{d}^{L+m}}{\rm{d}x^{L+m}}(x^2-1)^L$

The question that follows is driving me insane as it forms part of a derivation of the Associated Legendre Functions Normalization Formula: ...
1
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0answers
23 views

Understanding the expansion of product notation.

I have a question regarding the expansion of product notation in the picture below. Equation 3.1 in the attached picture is ...
3
votes
3answers
53 views

Evaluate this infinite product: $\prod_{n=3}^{\infty} \left(\;1-\frac{4}{n^2}\;\right)$

$$\prod_{n=3}^{\infty} \left(\;1-\frac{4}{n^2}\;\right)\;=\;\text{???}$$ I took the LCM and split the numerator as $(n+2)(n-2)$ and then took the product of the numerator and the denominator ...
-2
votes
3answers
46 views

Is there another notation for this set-theoretic formula? [closed]

I am writing a book. In my drafts there are formulas like $\prod_{X\in S} X$ or $\{ \operatorname{im} P \mid P\in\prod_{X\in S} X \}$. If there are other way to write the same expressions, I should ...
0
votes
1answer
35 views

Proof of Jacobi triple product by taking the limit

Assume that we know $$ \prod_{k=1}^{n}(1+q^{2k-1}z)(1+q^{2k-1}z^{-1})=C_{0}+\sum_{k=1}^{n}C_{k}(z^k+z^{-k}), $$ with $$ ...
4
votes
1answer
75 views

Euler Product formula for Riemann zeta function proof

In class we introduced Reimann Zeta function $$ \zeta (x)=\sum_{n=1}^{+\infty} \frac{1}{n^x} $$ And we proved its domain was $D=(1,+\infty)$ Now Euler proved that $$ \zeta(x)=\prod_{p\text{ ...
10
votes
3answers
319 views

Product of all real numbers in a given interval $[n,m]$

READ-ME I have now what I can call for myself answers to all my problems and subquestions proposed in this post, thus I accepted Strings answer as the answer to this question since it was of most ...
1
vote
2answers
71 views

Proving a result in infinite products.

We assume that $\sum |a_n|^{2}$ converges, then I want to conclude that $\prod (1+a_n)$ converges to a non zero element $\iff$ the series $\sum a_n$ converges. My attempt If $\prod (1+a_n)$ ...
2
votes
1answer
106 views

Prove that $\lim_{\ r\ \to \ \infty} \dfrac{r! r^x}{x(x+1)(x+2) \dots (x+r)} = \int_{0}^{\infty} t^{x-1} e^{-t} dt $

From Havil & Dyson, "Gamma: Exploring Euler's Constant", section 6.1 I can't prove the following Euler's theorem : ... on 13 October 1729, Euler had already proposed to Goldbach the ...
1
vote
4answers
139 views

If $a_n=\left(1-\frac{1}{\sqrt{2}}\right)\ldots\left(1-\frac{1}{\sqrt{n+1}}\right)$ then $\lim_{n\to\infty}a_n=?$

I have an objective type question:- If $$a_n=\left(1-\frac{1}{\sqrt{2}}\right)\ldots\left(1-\frac{1}{\sqrt{n+1}}\right)$$ then $\lim_{n\to\infty}a_n=?$:- A)$0$ B)limit does not exist ...
1
vote
1answer
48 views

Show $\cos \pi z=\prod\limits_{-\infty}^\infty \left(1-\frac{2z}{2n-1}\right) e^{\frac{2z}{2n-1}}$

I have already shown $\cos \pi z=\prod\limits_{n=1}^{\infty}\left(1-\frac{4z^2}{(2n-1)^2}\right)$. Here is what I have after that, \begin{equation*}\begin{split} \cos(\pi z) & = ...
31
votes
3answers
689 views

Interesting representation of $e^x$

So I discovered the following formula by using the Taylor series for $\ln (x+1)$ $$x= \ln ...
2
votes
3answers
181 views

What is $2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty$ equal to?

I came across this question while doing my homework: $$\Large 2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty=?$$ $$\small\text{OR}$$ ...
0
votes
0answers
35 views

Infinite expansion of difference of squares?

I was just wondering if the following is legitimate or if there's a problem with it: Using difference of squares, $a^n - b^n = (a^{n/2}-b^{n/2})(a^{n/2}+b^{n/2}) = ...
0
votes
1answer
10 views

How to prove the product representation of Baenes G-function

How to prove this formula of Barnes G-dunction $$G(z+1)=(2\pi)^{z/2}\text{exp}\left(-\frac{z(z+1)}{2}- \frac{\gamma z^{2}}{2}\right)\, ...
1
vote
3answers
35 views

Rearranging infinite product

I know that $$\frac{\sin x}x=\prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right).$$ Why exactly can I take the product and factor $x^2$? $$\prod_{n=1}^\infty ...
1
vote
0answers
25 views

Zeroes of infinite product function?

Given an entire function $\prod_{n=0}^∞ E_n (\frac{z}{z_n})$ , show that $(z_n )_n∈N$ is a complete list of the zeroes of this function in which each zero appears as many times as its multiplicity. ...
6
votes
1answer
284 views

What is the infinite product of (primes^2+1)/(primes^2-1)?

I have shown that the infinite product $$\prod_{p \in \mathcal{P}}\frac{p^2+1}{p^2-1}$$ is equal to $\frac{5}{2}$ (pretty remarkable!). I have checked this numerically with Wolfram Alpha for up to ...
0
votes
0answers
25 views

Uniform convergence of product

If I know that f is Lipschitz continuous, in particular $|f(2^{-j} x)-f(0)| \le L 2^{-j}|\xi|$ where $f(0)=1$. Does this help me to show that $$\Pi_{j=1}^{\infty} f(2^{-j}(x))$$ converges uniformly ...
1
vote
1answer
185 views

Proof that these two expressions are equivalent

I'm looking for algebraic proof that $$ y=\lim_{N \to \infty}\frac{\prod\limits_{n=0}^{N}x-n}{\prod\limits_{n=0}^{N-\lfloor x\rfloor}{x-\lfloor ...
1
vote
0answers
32 views

Constructing a smooth function whose roots consist only of each of the primes.

My first attempt: $$f(x) = \prod_{i=1}^\infty \left(1- \frac x {p_i} \right)$$ If we take a look at the Riemann zeta function: $$ \zeta(s) = \sum_{n = 1}^\infty \frac 1 {n^s} = \prod_{i = 0}^\infty ...
1
vote
4answers
74 views

Complex Finite Product $\prod_{k=0}^{n-1} (1-\zeta^k z)$

I am working on a review for a graduate level Complex Analysis course. The following problem is on the review: Let $\zeta= e^{\frac{2\pi i}{n}}$ $(n\in \mathbb{N})$; show that ...
0
votes
1answer
28 views

If $\prod _{n = 1 }^{\infty} a_n = 0$, then $lim_{n \rightarrow \infty} a_n = 0$ or $\exists n$ : $a_n =0$?

I need to prove above(or contraposition) using $\epsilon - \delta$ definition. How can I do? If $\prod _{n = 1 }^{\infty} a_n = 0$, then $lim_{n \rightarrow \infty} a_n = 0$ or $\exists n$ : $a_n ...
8
votes
2answers
114 views

Find $\lim\limits_{n\to\infty}{\left(1+\frac{1}{n^k}\right)\left(1+\frac{2}{n^k}\right)\cdots\left(1+\frac{n}{n^k}\right) }$ for $k=1, 2, 3, \cdots$

First of all, I already searched Google, math.stackexchange.com... I know $$ \lim_{n\rightarrow\infty} \left( 1+ \frac{1}{n} \right) ^n=e$$ That is $$ \lim_{n\rightarrow\infty} ...
6
votes
5answers
579 views

What is the limit of (Odd Numbers Product) / (Even Number Product)

What is the answer of this limit? $$ \frac{1\times3\times5\times\cdots}{2\times4\times6\times8\times\cdots} $$ I remember that it was something involving $\pi$. How can I compute it? in ...
5
votes
0answers
61 views

Closed form for the infinite product $\prod_{k=0}^{\infty} \left( 1-x^{2^k} \right)$

There is a known identity: $$\prod_{k=0}^{\infty} \left( 1+x^{2^k} \right)=\frac{1}{1-x}, ~~~~~|x|<1$$ It's easy to derive it by converting it to a telescoping product as shown in this answer. ...
3
votes
1answer
47 views

Closed form for $\prod_{n=0}^\infty (1-z^{2^n})$

Is there a closed form for the product $$f(z) = \prod_{n=0}^\infty 1-z^{2^n}$$ either as a formal power series or as an analytic function in the disk $|z| < 1$? It's not hard to see that Taylor ...
5
votes
1answer
71 views

Simple questions about infinite products

We just learned about infinite products in class. There's no textbook for the course so I am struggling with the following two basic problems. Let $ a_n(z) = 1 + b_n(z), |b_n(z)| \leq \lambda < 1, ...
0
votes
1answer
49 views

Uniform and absolute convergence of infinite trigonometric product

This is an exercise given to us in our analytic number theory class: Prove that $ \prod \cos(\tfrac{z}{2^n}) $ is uniformly and absolutely convergent on every closed disk $ \{ |z| \leq R \} $, hence ...
0
votes
2answers
74 views

Examples where series converges but product diverges and vice versa

Our professor gives us the following ungraded exercise for our analytic number theory class: Let $ E $ be a set with one element. Suppose $ (b_n) $ is a sequence with $ |b_n| \leq \lambda < 1 $, ...
3
votes
2answers
60 views

Infinite Earring and Product Space Homeomorphic

Old qual question here: We define two topological spaces $X$ and $Y$ as subspaces of certain topological spaces. $X$ is defined as a subspace of $\mathbb{R}^2$ which is the union of the infinite ...
3
votes
1answer
89 views

Euler's Basel problem proof explanation

First question: After using the function's Maclaurin series, we factor it $$\frac{\sin x}x=\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n}}{(2n+1)!}=\prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi} ...
5
votes
0answers
50 views

Can one define an uncountable infinite product? [duplicate]

The continuous version of a sum is commonly called an integral. But what would be the continuous version of a product? The same question in "pictures": $$ \sum \rightarrow \int $$ $$ \prod \rightarrow ...
3
votes
0answers
66 views

Infinite products of even analytic functions - highly accurate approximation

I discovered a way to evaluate infinite products of even analytic functions with high accuracy. $$ \prod_{k=1}^{\infty} f(k^2) \approx \prod_{k=1}^{\infty} ...
3
votes
1answer
42 views

How to prove that this infinite product of continued fractions converges to $1-\frac{1}{z}$?

$$\cfrac{z}{1+z} \cdot \cfrac{z}{1+z-\cfrac{z}{1+z}} \cdot \cfrac{z}{1+z-\cfrac{z}{1+z-\cfrac{z}{1+z}}} \cdots= 1-\frac{1}{z}$$ I propose that this works for any $z \in C$ if and only if $|z|>1$. ...
15
votes
3answers
399 views

Evaluate the infinite product $\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} \cdots$

I can show the convergence of the following infinite product and some bounds for it: $$\prod_{k \geq 2}\sqrt[k]{1+\frac{1}{k}}=\sqrt{1+\frac{1}{2}} \sqrt[3]{1+\frac{1}{3}} \sqrt[4]{1+\frac{1}{4}} ...
4
votes
3answers
88 views

Asymptotic formula for $\prod_{k=1}^{\infty}\zeta (2kn)$

Suppose $n\geq 1$ is a positive integer. Can we find an asymptotic formula for this product below. $$\prod_{k=1}^{\infty}\zeta (2kn)=\zeta (2n)\zeta (4n)\zeta (6n) \cdots$$ I tried to use $\zeta ...
3
votes
2answers
86 views

Prove these identities using Jacobi's triple product identity.

I am requesting help with deriving some identities from Jacobi's triple product identity: $$\sum_{n=-\infty}^{\infty}z^nq^{n^2}=\prod_{n\geq 0}(1-q^{2n+2})(1+zq^{2n+1})(1+z^{-1}q^{2n+1})$$ Here is ...
2
votes
0answers
33 views

Have I found the only solution to this curious-looking DE? $f[f(x+1)]=\prod_{i=0}^{\infty} f\left[ \frac{f^{(k)}(x)}{k!} \right]$?

This was just a curiosity I discovered while fooling around with the product function: One solution to the (admittedly weird looking) DE \begin{equation*} f[f(x+1)]=\prod_{k=0}^{\infty} f\left[ ...
3
votes
0answers
20 views

Convergence in the product of spaces of iteratively composed functions.

My question is a bit odd, in fact conceptually it is not difficult, only that it operates on objects that are complex (to me). I would like to check two types of convergence in the product of the ...
2
votes
1answer
35 views

Proving the equality of two infinite products

This question is essentially the same as this earlier question of mine. However, since Jacobi theta functions might be misleading in the title, in the following I just ask the exact question without ...