Mathematics

Questions tagged [diophantine-approximation]

For questions about approximating real numbers by rational numbers.

541 questions
129
votes
1 answer

Motivation of irrationality measure

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + \frac{x^n}{n!}+ \cdots$$ So the series for $e^{-1}$ is $$s_n…
Damien
  • 1
106
votes
20 answers

Can the golden ratio accurately be expressed in terms of $e$ and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the infamous values, $\large\pi$ and $\large e$ The closest…
Nick
  • 7,074
75
votes
6 answers

Proving that $m+n\sqrt{2}$ is dense in $\mathbb R$

I am having trouble proving the statement: Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $\epsilon > 0$ , the intersection of $S$ and $(0, \epsilon)$ is nonempty.
user11135
  • 863
69
votes
4 answers

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} \leqslant \sum_{n=1}^{\infty} \frac{1}{n^{1…
user55114
53
votes
1 answer

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that $$\frac{1}{2}=\inf\left\{\alpha\in\mathbb{R}^+:\sum_{n\geq 1}\frac{\sin(n^2)}{n^\alpha}\mbox{…
Jack D'Aurizio
  • 372,937
  • 42
  • 421
  • 888
46
votes
4 answers

Does the sequence $n+\tan(n), n \in\mathbb{N}$ have a lower bound?

Is the sequence $n+\tan(n), n \in\mathbb{N}$ bounded below? Intuitively I think it is not bounded below, but I have no idea how to prove it. It is like a Diophantine approximation problem, but most theorems seem to be too weak.
MathEric
  • 1,999
43
votes
5 answers

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated series The Simpsons. Thanks to Andrew Wiles, we all…
Stefan4024
  • 36,461
42
votes
1 answer

$\pi^4 + \pi^5 \approx e^6$ is anything special going on here?

Saw it in the news : $$(\pi^4 + \pi^5)^{\Large\frac16} \approx 2.71828180861$$ Is this just pigeon-hole? DISCUSSION: counterfeit $e$ using $\pi$'s Given enough integers and $\pi$'s we can approximate just about any number. In formal mathematical…
cactus314
  • 25,235
39
votes
5 answers

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). This question was inspired by the recently…
marty cohen
  • 110,916
37
votes
2 answers

Prove $\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}$

$$\sum_{n=1}^\infty\frac{\cot(\pi n\sqrt{61})}{n^3}=-\frac{16793\pi^3}{45660\sqrt{61}}.$$ Prove it converges and, evaluate the series. For the first part of the question, I prove it converges by considering the irrationality measure, $$|\sin…
Tianlalu
  • 5,417
33
votes
1 answer

Irrationality of $\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$

Let $\mathbb{P}$ be the set of prime numbers, and consider $m=\displaystyle\sum_{p\in\mathbb{P}} \frac{1}{2^{p}}$. Is $m$ irrational? In the following paper , the author recalls several sufficient criteria for irrationality. When applying some of…
Chazz
  • 889
  • 6
  • 18
32
votes
1 answer

Examples of transcendental functions giving almost integers

Informally speaking, an "almost integer" is a real number very close to an integer. There are some known ways to construct such examples in a systematic way. One is through the use of certain algebraic numbers called Pisot numbers. These numbers…
user346361
30
votes
3 answers

Is the sequence $(B_n)_{n \in \Bbb{N}}$ unbounded, where $B_n := \sum_{k=1}^n\mathrm{sgn}(\sin(k))$?

This question is kind of an extension of a previous question I asked here . The infinite series $$\sum\frac{\mathrm{sgn}(\sin(n))}{n}$$ does converge, but I would like to know if Dirichlet's test can be used to prove the convergence with…
Jaeseop Ahn
  • 679
27
votes
6 answers

If $\frac{m}{n}$ is a good approximation to $\sqrt{2}$ then $\frac{m+2n}{m+n}$ is a better one

I'm a first year Undergraduate student from India. Our professor is going to start a Real Analysis course in September and I was preparing for the initials. I tried and solved many problems, but this one has me confused. Probably the main reason for…
Gaurav Tiwari
  • 1,579
26
votes
2 answers

Is there any real number except 1 which is equal to its own irrationality measure?

Is there any real number except $1$ which is equal to its own irrationality measure ? If so, then what is the cardinality of the set of all such numbers? Is the set dense on any interval? Is it measurable?
Vladimir Reshetnikov
  • 32,880
1
2 3
…
36 37