6
votes
4answers
200 views

What's the value of $n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}$ for $n\in\mathbb{C}$?

Write $$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}}$$ so that $\phi_n=n+\frac{n}{\phi_n}$, which gives $\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}$. We know ...
17
votes
2answers
503 views

Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 ...
1
vote
2answers
80 views

Reconstruct a quadratic irrational from its periodic continued fraction

How can one find a quadratic irrational when knowing its periodic continued fraction? For example(using Wikipedia notion), how can one find the quadratic irrational that its continued fraction is ...
0
votes
0answers
68 views

Proof of a striking identity of Tito Piezas III

In the q series blog of Tito Piezas here . He gives a very striking relation I am wondering on how to prove that ?
1
vote
1answer
58 views

Solve the equation $285x \equiv 177 \pmod{924}$ using continued fraction

Solve the equation $285x \equiv 177 \pmod{924}$ using continued fraction My attempt(using Wikipedia notion): Continued fraction form for $\frac{924}{285}$ is $[3;4,6,1,9]=[q_1;q_2,q_3,q_4,q_5]$ ...
0
votes
0answers
21 views

uniform spanning tree of $2 \times n$ graph

In Probability on Trees and Networks Chapter 1 study the uniform spanning tree on the ladder graph: _ |_| |_| |_| ... |_| |_| The probability the bottom rung ...
1
vote
0answers
57 views

Is this a bounded sequence ? (about continued fraction)

Represent $\sqrt{2}$ in the form $$\sqrt{2}=1+\frac{8}{A_1+\displaystyle\frac{8}{A_2+\displaystyle\frac{8}{A_3+\ddots}}},$$ where $A_n$ is a positive integer and $A_n \geq 8$ for all $n$. So we have ...
5
votes
1answer
56 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 : ...
1
vote
1answer
58 views

How to prove continued fraction convergents of a number

Let $x=1+\sqrt{3}$. Prove that in pairs the continued fraction convergents of $x$ are $a_n$/$b_n$ < x < $c_n$/$d_n$ where $a_1$ = 2, $b_1$ = 1, $c_1$ = 3, $d_1$ = 1, $a_{n+1}$ = 2$c_n$ + $a_n$, ...
1
vote
1answer
43 views

Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show that ...
1
vote
1answer
21 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
1
vote
1answer
28 views

How to restrict the output values of a continued fraction?

I understand that a continued fraction of the form: $g(n_1,n_2,n_3,n_4,n_5,\ldots)= n_1 + \cfrac{1}{n_2 + \cfrac{1}{n_3 + \cfrac{1}{n_4 + \cfrac{1}{n_5+\cdots} } } }$ gives a unique irrational number ...
1
vote
2answers
111 views

Infinite continued fraction expansion

How can we find the first six partial quotients of the infinite continued fraction expansion of $\sqrt[3]2$? I know how to do this by expanding when we have a square root function... but I"m not sure ...
1
vote
1answer
106 views

Pell's Equation through Continued Fractions

Use continued fractions to find the minimal solution to $x^2-11y^2=1$. I know that $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+...}}}$ I took $\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\sqrt{11}}}$ and I ...
0
votes
0answers
40 views

Intermediate convergents of a continued fraction.

I have been studying continued fractions and convergent's properties, and i have a questions about "intermediate convergents" I have read that the expression of the intermediate convergents (those ...
2
votes
1answer
72 views

About continued fractions as best rational approximations

I'm reading this notes about continued fractions: http://www.math.jacobs-university.de/timorin/PM/continued_fractions.pdf I had no problems understanding everything there, except one thing that has ...
9
votes
0answers
137 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
3
votes
1answer
86 views

Inequality related to the continued fraction expansion of sqrt(3)

I am working on a problem related to the continued fraction expansion of $\sqrt3$. If $p_k$ and $q_k$ denote the numerator and denominator, respectively, of the $k$th convergent, I should show that ...
8
votes
2answers
109 views

Reference for relation between class number of $\Bbb Q[\sqrt{-p}]$ and partial quotients of $\sqrt p$

So in Ireland and Rosen's, "Classical Introduction to Modern Number Theory", they mention the following incredible fact at the end of Chapter 13, section 1. Suppose $p \neq 3$ and $p \equiv 3 \pmod 4$ ...
2
votes
1answer
85 views

Pell-like equations and continued fractions

Why does the continued fraction method work? Could be applied in order to solve, for example, $x^{17}-19y^{17}=1$ ?
4
votes
2answers
115 views

A numerical coincidence with continued fractions

My brother built a garage that measures 45 feet by 30 feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches, $$ ...
0
votes
0answers
82 views

By establishing a recurrence relation and using induction, or other-wise, show that this sequence is 3-adically Cauchy?

this is a question from a book I'm struggling with, please could you provide a clear proof Consider the sequence of rational numbers $a_1 = 1+3,a_2 = 1+\frac{3}{1+3},a_3= 1 + \cfrac{3}{1 ...
5
votes
5answers
246 views

Continued Fraction [1,1,1,…]

If the continued fractional representation of an irrational number $\alpha$ is given by [1,1,1,...], I can compute that $\alpha = \frac{1+\sqrt{5}}{2}$ by solving the equation $\alpha = 1+ ...
13
votes
2answers
313 views

How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
3
votes
4answers
2k views

Continued fraction of a square root

If I want to find the continued fraction of $\sqrt{n}$ how do I know which number to use for $a_0$? Is there a way to do it without using a calculator or anything like that? What's the general ...
8
votes
1answer
132 views

optimality of 2 in a continued fraction theorem

I'm giving some lectures on continued fractions to high school and college students, and I discussed the standard theorem that, for a real number $\alpha$ and integers $p$ and $q$ with $q \not= 0$, if ...
2
votes
1answer
236 views

Convergent fraction for constant $e$?

I've just learned about e. I am very much the novice and my problem is that while trying to calculate the convergent fractions for e. For instance: ...
3
votes
3answers
185 views

Solution to $x=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ldots}}}$ [duplicate]

Possible Duplicate: Why does this process, when iterated, tend towards a certain number? (the golden ratio?) Please post your favorit solution to the following Compute ...
38
votes
1answer
676 views

A new continued fraction for Apéry's constant, $\zeta(3)$?

As a background, Ramanujan also gave a continued fraction for $\zeta(3)$ as $\zeta(3) = 1+\cfrac{1}{u_1+\cfrac{1^3}{1+\cfrac{1^3}{u_2+\cfrac{2^3}{1+\cfrac{2^3}{u_3 + \ddots}}}}}\tag{1}$ where the ...
7
votes
2answers
239 views

Continued fractions

I'd really love with concluding that for given integers $a_0,a_1,...a_N$ with $a_i>0$ for $i>0$, representing the continued fraction $[a_0; a_1,....,a_N]$, with the following recursion: ...
0
votes
1answer
68 views

Continued Fraction: Why do we get with $\gamma \in \mathbb{R}\setminus \mathbb{Q}$ the CF $\frac{1}{\gamma}=\langle0;a_0,a_1,\dotsc\rangle$

I have a question concerning continued fractions: If we have $\gamma \in \mathbb{R} \setminus \mathbb{Q}$ and $\gamma=\langle a_0;a_1,a_2,\dotsc\rangle$. Why do we get $$\frac1\gamma = \langle ...
2
votes
1answer
128 views

Reciprocal of a continued fraction

I have to prove the following: Let $\alpha=[a_0;a_1,a_2,...,a_n]$ and $\alpha>0$, then $\dfrac1{\alpha}=[0;a_0,a_1,...,a_n]$ I started with ...
2
votes
1answer
83 views

Show $p(n)=n(p_{n-1}+p_{n-2})+(n-1)p_{n-3}+(n-2)p_{n-4}+…+3p_1+2p_0+2$

I have to show the following: Let $N_k=\frac{p_k}{q_k}$ with $\alpha=\langle 1;2,3,4,...,n,n+1\rangle$ and $n \in \mathbb{N}$. Then $\forall n \in \mathbb{N}$ with $n\geq 3$, ...
3
votes
0answers
184 views

Approximation of a real number as a linear combination of two reals with coprime integral coefficients

Given two nonzero real numbers $x$ and $y$ such that $y/x$ is irrational, a real number $z$ to be approximated, and a tolerance $\epsilon$, give me an algorithm that will provide coprime integers $a$ ...
0
votes
3answers
1k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) ...
16
votes
3answers
879 views

Motivation behind this eccentric Ramanujan Identity

I just visited the MathJaX page due to the Math.SE website showing some problems while loading the page. I saw some demo math equations samples at this page, when this identity actually caught my ...
6
votes
2answers
202 views

A question on continued fraction

Let $a$ be a positive irrational number. Let $p_k/q_k, p_{k+1}/q_{k+1}$ be two consecutive convergents of its simple continued fraction, where $k\ge 1$. Is it possible that both ...
7
votes
0answers
431 views

What can Euler's identity teach us about (generalised) continued fractions?

We know that $$e^{i \pi} = -1 .$$ We can transform all of the components of this identity into (generalized) continued fractions. When we start of with $\pi$, we see that $$ \Big(3+ ...
10
votes
1answer
489 views

Continued Fraction expansion of tan(1)

Prove that continued fraction of tan(1)=[1;1,1,3,1,5,1,7,1,9,1,11,...]. I tried using the same sort of trick used for finding continued fractions of quadratic irrationals and trying to find a ...
4
votes
2answers
302 views

Adding integers to an infinite continued fraction expansion doesn't change the value?

I'm learning about continued fractions, and I've enjoyed them so far, but I'm unsure if I've done the following correctly. I have no real experience with analysis, so I'm not sure if my reasoning is ...
5
votes
1answer
413 views

Continued Fraction of an Infinite Sum

What is the continued fraction for $\displaystyle\sum_{i=1}^n\frac{1}{2^{2^i}}$ It seems to be "almost" periodic, but I can't figure out the exact way to express it.
7
votes
3answers
658 views

Why are some mathematical constants irrational by their continued fraction while others aren't?

Catalan's Constant and quite a few other mathematical constants are known to have an infinite continued fraction (see the bottom of that webpage). On wikipedia (I'm sorry, I can't post anymore ...
10
votes
2answers
869 views

How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction?

Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions. Description of the Bauer-Muir transformation ...
5
votes
1answer
681 views

A nicer proof of Lagrange's 'best approximations' law?

Let $p_N/q_N$ be the $N^\text{th}$ convergent of the continued fraction for some irrational number $\alpha$. It turns out that for any other approximation $p/q$ (with $q \le q_N$) which isn't a ...