For questions about approximating real numbers by rational numbers.

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29
votes
4answers
3k views

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

I am having trouble proving the statement: Let $S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$. Prove for every $\epsilon > 0$, The intersection of $S$ and $(0, \epsilon)$ is nonempty.
9
votes
3answers
576 views

A series to prove $\frac{22}{7}-\pi>0$

After T. Piezas answered Is there a series to show $22\pi^4>2143\,$? a natural question is Is there a series that proves $\frac{22}{7}-\pi>0$? One such series may be found combining ...
1
vote
1answer
237 views

Series and integrals for inequalities and approximations to $\pi$

Fundamentals Two beautiful expressions that relate $\pi$ to its convergents are Dalzell integral $$\frac{22}{7}-\pi=\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx$$ (see Why do we need an integral to prove ...
43
votes
3answers
2k views

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 + ...
14
votes
1answer
612 views

Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
2
votes
2answers
172 views

How find the value $\beta$ such $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Find all positive real number $\beta$,there are infinitely many relatively prime integers $(p,q)$ such that $$\left|\dfrac{p}{q}-\sqrt{2}\right|<\dfrac{\beta}{q^2}$$ maybe this problem background ...
75
votes
14answers
10k views

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 ...
11
votes
2answers
348 views

How was this approximation of $\pi$ involving $\sqrt{5}$ arrived at?

The Wikipedia article for Approximations of $\pi$ contains this little gem: $$ \pi \approx \frac{63}{25}\times\frac{17 + 15\sqrt{5}}{7 + 15\sqrt{5}} $$ which is clearly in $\mathbb{Q[\sqrt{5}]}$. ...
13
votes
3answers
879 views

Calculate $\lim_{n \to \infty} \sqrt[n]{|\sin n|}$

I am having trouble calculating the following limit: $$\lim_{n \to \infty} \sqrt[n]{|\sin n|}\ .$$
20
votes
5answers
5k views

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 ...
4
votes
0answers
153 views

Is an integer in the interval $\left[\small {3^n+1 \over 2^n+1}\lt {3^n \over 2^n} \lt{3^n-1 \over 2^n-1}\right] $ for some $\small n>1$?

Consider the expression $$ f_n(j)= {3^n+j \over 2^n+j} $$ where we select some fixed $n \gt 1$ and let j vary over the reals from +1 to -1 . I'm concerned with the problem, whether in the interval ...
2
votes
1answer
114 views

Series and integrals for inequalities and approximations to $\log(n)$

The following series and integrals relate log(2) to its third and fourth convergents, $\frac{2}{3}$ and $\frac{7}{10}$. $$\begin{align} \log\left(2\right)-\frac{2}{3} &= \sum_{k=1}^\infty ...
1
vote
1answer
80 views

Gap between smooth integers tends to infinity (Stoermer-type result)?

Consider the following claim : (*) Let $P$ be a finite set of primes, let $S$ be the set of natural numbers all of whose divisors are in $P$, and let $s_n$ denote the $n$-th element of $S$. Then ...
23
votes
2answers
874 views

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 ...
15
votes
0answers
409 views

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 ...
10
votes
1answer
429 views

For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?

Disclaimer: The original version of this question focused on $2^n$ in lieu of $n^2$. It is in the hope that the question is easier with $n^2$ that I changed it. I have an always-nonnegative (on the ...
7
votes
5answers
788 views

Is π unusually close to 7920/2521?

EDIT: One can look at a particular type of approximation to $\pi$ based on comparing radians to degrees. If you try to approximate $\pi$ by fractions of the form $180n/(360k+1)$, you can find that ...
11
votes
1answer
202 views

What is the sum of the squares of the differences of consecutive element of a Farey Sequence

A Farey sequence of order $n$ is a list of the rational numbers between 0 and 1 inclusive whose denominator is less than or equal to $n$. For example $F_6= ...
6
votes
1answer
162 views

Finding near-integers in a range

I have a (transcendental) constant $\alpha$ and a fixed parameter $\varepsilon>0.$ I'd like to find all positive integers $n<\ell$ for which $\|n\alpha\|<\varepsilon,$ where $\|x\|$ is the ...
5
votes
1answer
115 views

Intuition behind proof in the Rudin book that there is no largest/smallest real number. [duplicate]

In Rudin's Principles of Mathematical Analysis (3rd ed), he proves (at the very beginning: example 1.1) that the set $A$ of all positive rationals $p$ such that $p^2<2$ contains no largest number ...
5
votes
2answers
78 views

Closeness of $n! \ x$ to integers for irrational $x$

This question came up in the comments to another question. Is there an irrational number $x$ such that, for sufficiently large $n$, the product $$ n! \ x $$ is arbitrarily close to an integer? More ...
4
votes
1answer
109 views

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits.

Prove that powers of any fixed prime $p$ contain arbitrarily many consecutive equal digits. It is an intuitive re-statement of Baltic Way 2012 (I think there are shortlists in Baltic Way every ...
2
votes
1answer
188 views

Existence of a simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and for every ...
2
votes
2answers
325 views

Does every normal number have irrationality measure $2$?

A normal number is a number whose digit expansion in any base is "uniform" in the sense that all finite digit strings occur with the "statistically expected" frequency. I read a sentence somewhere ...
12
votes
3answers
576 views

When is a sequence $(x_n) \subset [0,1]$ dense in $[0,1]$?

Weyl's criterion states that a sequence $(x_n) \subset [0,1]$ is equidistributed if and only if $$\lim_{n \to \infty} \frac{1}{n}\sum_{j = 0}^{n-1}e^{2\pi i \ell x_j} = 0$$ for ever non-zero integer ...
8
votes
1answer
112 views

Near-integer solutions to $y=x(1+2\sqrt{2})$ given by sequence - why?

EDIT: I've asked the same basic question in its more progressed state. If that one gets answered, I'll probably accept the answer given below (although I'm uncertain of whether or not this is the ...
1
vote
8answers
231 views

Rational number that approximates $\sqrt{3}$

Questions: Show that is is theoretically possible to find a rational number that approximates $\sqrt{3}$ with an error less than $0.001$. Explain how you would go about determining a ...
4
votes
1answer
64 views

Proving that sequence elements satisfy inequality involving $\mod{1}$

I'm trying to prove that $n(3-\sqrt{8}) \; (\!\!\! \mod{1}) < m(3-\sqrt{8}) \; (\!\!\! \mod{1})\;\; \forall \; m < n \;|\;n,m \in \mathbb{N}$ $\leftrightarrow n \in ...
0
votes
2answers
266 views

How find all positive real $\beta$ such A finite number of $\left|\frac{p}{q}-\sqrt{2}\right|<\frac{\beta}{q^2}$

Define sequence $\{a_{n}\}$,such $$a_{1}=1,a_{2}=2,a_{k+2}=2a_{k+1}+a_{k},k\ge 1$$ Find all positive real number $\beta$,such only have a finite number of relatively prime integers $(p,q)$ ...
0
votes
0answers
121 views

integer solutions to $a^m+nx^2 = y^n$ with various conditions

I consider the following equation with conditions of obtaining solutions $$a^m+nx^2 = y^n$$ This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = ...
-3
votes
2answers
149 views

A series of positive terms to prove $\pi>\frac{333}{106}$

This is a consequence of the answer to that question. A proof that $\pi > \frac{333}{106}$ is given by the series of positive terms $$\pi-\frac{333}{106} \\ =\frac{48}{371} \sum_{k=0}^\infty ...