# Tagged Questions

166 views

### Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
118 views

### What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental. My question is what are the transcendentals ...
60 views

### Transcendentals as the Roots of Infinite Polynomials

I have always been taught that the difference between an algebraic and a transcendental number is that the former is the root to a polynomial of ${\bf finite}$ degree with integer coefficients. I did ...
107 views

### Prove of trancendence of $\ln(2)$.

Where can I find some proofs for another transcendental numbers, like Hermite/Lindemann theorem proofs for $e/\pi$? For instance, prove that $\zeta(3)/\ln(2)$ is a transcendental number.
45 views

### What are some algorithms that can be used to test if a number is transcendental or not?

Well according to the definition of transcendental numbers I find that its any number that doesn't have any polynomial equation of any degree with integer coefficients summing up to 0. So ...
66 views

### The smallest class of numbers closed under addition, multiplication, and exponentiation

Let $\def\A{\mathfrak A}\A$ be the smallest subset of $\Bbb C$ that contains the algebraic numbers and also all numbers of the form $$\sum \alpha_i^{\beta_i}$$ where the $\alpha_i, \beta_i$ are ...
130 views

### Does the number $2.3\,5\,7\,11\,13\ldots$ exist and, if so, is it rational or irrational &/or transcendental? [duplicate]

Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$ if so, will it be rational or irrational &/or transcendental?
139 views

### Uncountable sets of transcendental numbers

As a sort of follow up question to a previous question found here, besides the Liouville numbers, are there any other uncountable collections of transcendental numbers that are known? Clearly you ...
87 views

### Transcendental numbers involving primes?

Is the prime zeta function value $$P(2)=\sum_{p \in \mathrm{primes}} \frac{1}{p^2} = 0.452247420041065498506543364832247934173231343\ldots$$ a transcendental number ? What about the following sum ...
42 views

### Fast converging sums involving tetrations

Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check. Now my question : ...
64 views

### Number made from the first digits of $2^n$

Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are : ...
145 views

### Irrationality/Transcendentality of values of $e^{e^x}$

1) Is $e^{e^x}$ irrational for all rational $x$? It is known that $e^x$ is transcendental for every nonzero algebraic $x$. But this dos not help here because for transcedental $x$, $e^x$ can be ...
444 views

373 views

### Certain Liouville Numbers

A Liouville number is a number which can be approximated very closely be a sequence of rational numbers (here is the rigorous definition I am working off of: ...