8
votes
2answers
162 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 ...
1
vote
0answers
41 views

Series of polynomials and uniformly convergence

It's part of the proof of a Lemma of an article I was reading (Algebraic values of transcendental functions at algebraic points). I couldn't understanding one thing: Let f be a complex function such ...
3
votes
2answers
92 views

A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
2
votes
1answer
44 views

$n^n$ cannot be expressed as a recurrence with polynomial coefficents

We say that a sequence $a(n)$ is $P$-recursive if there exist polynomials $p_0(n),\ldots,p_k(n) \in \mathbb{Q}[n]$ such that $$p_k(n) a(n+k) + \cdots p_0(n) a(n) = 0.$$ I would like to show that the ...
2
votes
1answer
78 views

Does the Thue-Morse sequence form a Sturmian Word?

Does the Thue-Morse sequence form a Sturmian Word? The Thue-Morse sequence 011010011001001..., formed by appending the negation of the existing string, yields the ...
17
votes
2answers
783 views

Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the ...
8
votes
0answers
95 views

On Bailey and Crandall's sum for $\sum_{n=0}^\infty \frac{1}{5^{5n}}\left(\frac{5}{5n+2}+\frac{1}{5n+3}\right)$

On page 20 of "On the Random Character of Fundamental Constant Expansions", Bailey and Crandall gave the rather unusual sum, $$u_2 = \sum_{n=0}^\infty ...
7
votes
1answer
184 views

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Transcendental?

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Clearly this series is convergent (compare to geometric series with ratio 1/2). I'm sure it's irrational since a rational number written in base 2 ...
11
votes
0answers
174 views

Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?

This question is inspired by my answer to the question "How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?". The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer ...
5
votes
2answers
84 views

For integer $k > 1$, is $\sum_{i=0}^{\infty} 1/k^{2^i}$ transcendental or algebraic, or unknown?

Title says it all, I have an itch about series like this that seem to fall in the gray area where classical proofs that rational partial sums that converge too quickly must converge to transcendental ...
1
vote
1answer
109 views

$\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$ Are there any known ...
4
votes
0answers
95 views

The Tribonacci constant and the Dragon

Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation, $$4^x(2^x-1)=(2^x+1)$$ Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
11
votes
1answer
316 views

Proving that $\frac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\frac{(-1)^{(p_{k}-1)/2}}{p_{k}} \right )^{-1}$ an identity of Euler's.

This is another identity of Euler's relating $\pi$ to the prime numbers, available here \begin{align*} \dfrac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\dfrac{(-1)^{\dfrac{p_{{k}}-1}{2}}}{p_{k}} \right ...
12
votes
2answers
425 views

Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$

Long time ago I've been playng with formulas for $\pi$ and found that one above in the title which can also be expressed as \begin{align*} ...
9
votes
2answers
251 views

Erdős: Sum of rational function of positive integers is either rational or transcendental

I am trying to find a conjecture apparently made by Erdős and Straus. I say apparently because I have had so much trouble finding anything information about it that I'm beginning to doubt its ...
4
votes
0answers
437 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
6
votes
3answers
287 views

What is $\sum\limits_{n=0}^{\infty} r^{an^2 + bn + c}$ ? or: is $0.0100100010000100001…$ transcendental?

The idea is a more convenient form for $N = 0.01001000100001000001...$ in base $r$, hopefully to show whether it is transcendental. Sorry for brevity.