For questions on continued fractions.

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2
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2answers
169 views

Question in fraction (not simple )

I have a question and its answer but I don't know how can i solve $$\frac {37}{13} = 2+ \frac {1}{x+\frac{1}{5+\frac{1}{y}}} $$ the answer $ x =1, y=2$ Could any one explain how to solve this ?? ...
7
votes
4answers
635 views

How to detect when continued fractions period terminates

I'm doing continued fractions arithmetic. Is there a method to detect when a continued fractions period terminates? Let me give you an example: $\sqrt{2} = [1; \overline{2}]$, $\sqrt{7} = [2; ...
3
votes
0answers
72 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
19
votes
2answers
625 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 ...
2
votes
1answer
61 views

Proving something about $|f(x)|$ when the lim of $f(x)/x^2$ is known

I've been trying to crack this issue for 2 days and I got pretty much nothing Given that $f$ is a continuous function and the following limits exists and are finite: $$ (1) ...
1
vote
1answer
53 views

Math and Taxes - Items versus Sum Total differs

First, The question is a bit confusing as I am not really sure how to word this problem as a question. The math problem I encountered which is a bit of an anomaly is this : Suppose you are ...
1
vote
1answer
34 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
1
vote
3answers
159 views

Convergents of square root of 2

On wikipedia I read about the continued fraction of the square root of 2: $$1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{...}}}}$$ The first convergents are ...
1
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1answer
56 views

Is there a bijection between real numbers and continued fractions?

It is known that there is a bijection between rational numbers and finite continued fractions, so every rational number is uniquelly identified by a finite continued fractions and vice versa. It is ...
1
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0answers
31 views

Champernowne constant - summation and behavior of terms in continued fraction expansion

Is there an infinite summation that gives the Champernowne constant? Wikipedia has one, and so does Wolfram MathWorld. Are they valid? If so, could someone explain why, i.e how do they work? Also, ...
6
votes
3answers
877 views

Computing Infinite Continued Fractions

I am looking for "tricks" used to compute infinite continued fractions. For example, $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots}}}$$ is the golden ratio since if we denote it by $x$, then we have ...
6
votes
1answer
91 views

Infinite Continued Fraction Notation

I can't find anywhere via googling; is there some sort of $\sum$ like notation for infinite continued fractions? In other words, for a sum we do this: $$ 1+x+x^2+x^3+... = \sum_{n=0}^\infty x^n $$ ...
2
votes
2answers
289 views

Taking the negative of a continued fraction

If I have a continued fraction for an irrational number $z= \langle a_0;a_1,a_2,a_3,\ldots\rangle$ it seems that $(-1)*z = \langle-a_0;-a_1,-a_2,-a_3,\ldots\rangle$. Is this true? In general, if you ...
1
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2answers
150 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
100 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
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1answer
73 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
2answers
92 views

Closed form for this continued fraction [closed]

Is there a closed form for this continued fraction? $$x+\frac{1}{x+\frac{1}{x+\frac{1}{...}}}$$
1
vote
1answer
83 views

Nested square root of continued fraction references

$$\sqrt {a_0 + \cfrac{b_1}{\sqrt{a_1 + \cfrac{b_2}{\sqrt{a_2 + \cfrac{b_3}{ \ddots }}}}}}$$ Are there any texts that explain how to deal with expressions like this?
1
vote
2answers
76 views

Find value of unending continued fraction

I am trying to find the limit if it exists for the following unending continued fraction: $$1+{1\over{2+{1\over2+{1\over{2+...}}}}}$$ I have discovered this is the continued fraction for $\sqrt2$, ...
1
vote
0answers
75 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 ...
3
votes
2answers
71 views

Why is the coefficient in front of $\sqrt n$ always 1 in the intermediate terms for finding the continued fraction expansion of $\sqrt n$?

After playing around on paper for a bit, I came up with a short python generator to find the continued fraction expansion of $\sqrt n$. I understand why it gets the right answer when it gets an ...
7
votes
5answers
730 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 ...
4
votes
0answers
102 views

Different ways of operating an infinite continued fraction

Given the continued fraction below, $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}+\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}} $$ I wanted to know to which number it converged, so I ...
8
votes
2answers
125 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 $p \equiv 3 \pmod 4$ ...
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0answers
32 views

Explanations of the Euler's continued fractions to compute exponential

After looking for explanations of the Euler's continued fractions to compute exponential on internet and after reading Euler's explanations about, I still don't understand how Euler find this ...
5
votes
1answer
68 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 : ...
9
votes
3answers
308 views

Value of $f'(0)$ if $f(x)=\frac{x}{1+\frac{x}{1+\frac{x}{1+\ddots}}}$

Consider the function $$f(x)=\cfrac{x}{1+\cfrac{x}{1+\cfrac{x}{1+\ddots}}} $$ Determine the value of $f'(0)$. I tried to differentiate $f(x)$ but it is not subject to chain rule, and now I'm stuck. ...
0
votes
0answers
89 views

Calculate an infinite continued fraction as special function

It is possible to convert this infinite continued fraction $\cfrac{1}{-a+\cfrac{b\;f(0)}{a+\cfrac{b\; f(1)}{-a+\cfrac{b\; f(2)}\ddots}}}$ to a special function ? Please, how do it? where : $(a,b) ...
12
votes
1answer
162 views

Evaluation of a slow continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
1
vote
1answer
71 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
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2answers
55 views

Short proof of Seidel-Stern theorem on continued fractions

Let $\mathbf{a}=\{a_n:n\ge0\}$ be a sequence of positive real numbers, and consider the formal continued fraction $$K(\mathbf{a})=a_1+\cfrac{1}{a_1+\cfrac{1}{a_2+\ddots.}}$$ Seidel-Stern Theorem. If ...
0
votes
0answers
90 views

Why is this finite continued fraction expression equal to $1$?

How find this value I use computer relsut is $1$, maybe this problem have the general relsut But How can prove this by hand? maybe have nice methods? Thank you
1
vote
1answer
57 views

Finite Continued Fraction Proof

Let $\alpha = \left[a_0, a_1, a_2,\cdots,a_n\right]$ be a finite continued fraction with $a_0 > 0$ and let $C_i = p_i/q_i$ be the convergent of $\alpha$. If $i\ge 1$, prove that ...
7
votes
1answer
130 views

Is this a misuse of the word “evaluate”?

I have found the following use of the word "evaluate" in several math books: "To evaluate the continued fraction, start at the bottom and work your way up:" $\huge \underbrace{2 + ...
1
vote
2answers
60 views

Continued fraction explanation

This is about simple infinite continued fraction. I don't understand the line '...then $C_0 < x < C_1$'. $C_k$ here refers to $C_k=[a_0;a_1,a_2,...,a_k]$ where $1 \leq k \leq n$. $C_o=a_0$. ...
1
vote
1answer
61 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 ...
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) ...
1
vote
1answer
25 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
40 views

Continued fraction with alternative signs

Let $a_n$ be a sequence of real numbers. We can define a formal finite continued fraction as usual ...
0
votes
1answer
73 views

Convergent Sequence from Introduction to Analysis

Consider the sequence of real numbers $$\frac 12, \cfrac 1{2+\cfrac 1 2}, \cfrac 1{2+\cfrac 1{2+\cfrac 12}}, \ldots.$$ Show that this sequence is convergent and find its limit by first ...
1
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1answer
31 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
154 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 ...
4
votes
1answer
64 views

Expressing $1+a_1(b_1+a_2(b_2+a_3(b_3+a_4(b_4+a_5(\cdots)))))$ as an infinite continued fraction.

Euler derived the following identity $$ 1+a_{1}+a_{1}a_{2}+a_{1}a_{2}a_{3}+\cdots= \cfrac{1}{ 1- \cfrac{a_{1}}{ 1+a_{1}- \cfrac{a_{2}}{ 1+a_{2}- \cfrac{a_{3}}{ ...
2
votes
0answers
83 views

Interesting Recursive Continued Fraction Limit

I was messing around with recursive functions the other day and came up with something that could be interesting: Definition of $\bar{\Xi}(n)$:\ Let $\Xi ...
0
votes
1answer
71 views

Continued fraction : How to find the first 3 terms

I can't calculate the exact first tree terms $F_0$, $F_1$ and $F_2$ of this continued fraction : $$F_n=\cfrac{1}{-\text{i$\omega $}\,+A\,\cfrac{(n+1)^2}{{4 (n+1)^2-1}}F_{n+1}}$$ $A$ and ...
10
votes
0answers
167 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
174 views

Definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$

Is there a definition of $ 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{\ddots}}}}$? I am somewhat familiar with continued fractions; that is, I am aware that their convergence depends on whether ...
2
votes
1answer
97 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 ...
5
votes
1answer
87 views

Is the infinite continued fraction $[0;0,0,\ldots]=0$?

Wolfram|Alpha states that the infinite continued fraction $$\cfrac{1}{0+\cfrac{1}{0+\cdots}}=0.$$ Assuming $[0;0,0,\ldots]$ exists implies that the continued fraction is $1$, since ...
3
votes
1answer
99 views

Are continued fractions a mere curiosity?

Does algebraic geometry have a good understanding of continued fractions? What kind of geometric or arithmetic information does a continued fraction expansion contain, if any? Are there rings of ...