7
votes
0answers
154 views

Continued fraction and double series.

From Euler's continued fraction formula, we have $$x = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}\,$$ and $$x = 1 + \sum_{i=1}^\infty r_1r_2\cdots r_i = ...
1
vote
0answers
33 views

Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
6
votes
2answers
179 views

Evaluation of a class of continued fractions

Is there a closed-form way of writing the continued fraction: $$ 1 + \frac{2}{3+ \frac{4}{5 + \frac{6}{7 + ...}}} $$ EDIT: Since the above has been determined as $\frac{1}{\sqrt{e}-1}$, is there a ...
6
votes
3answers
208 views

Where did the negative answer come from in $1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{\ddots }}}$?

In this question we increased solution domain by squaring both sides of equation but what about this one ? Here the question is to evaluate ...
0
votes
1answer
70 views

Convergent's numerators of the continued fraction for $\pi$

Call $C_{\pi} = \{ 1,3,22,333,355,… \}$, it´s the sequence of the numerators of convergents of the continued fraction for $\pi$, its OEIS' A002485, http://oeis.org/A002485. Let $n \in C_{\pi}$, such ...
16
votes
1answer
205 views

An infinite series plus a continued fraction by Ramanujan

Here is a famous problem posed by Ramanujan Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + ...
0
votes
1answer
74 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
3
votes
2answers
72 views

Evaluating $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } }$

What is the value of the fraction $\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $? I let $x=\frac{2}{3-\frac{2}{3-\frac{2}{3-\ldots} } } $ and hence $x=\frac{2}{3-x} $ which gives $x=1$ or $x=2$. ...
0
votes
0answers
88 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
0answers
69 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 ...
0
votes
1answer
68 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 ...
3
votes
1answer
48 views

Why does this pattern fail (sometimes) for the continued fraction convergents of $\sqrt{2}$?

This is connected to my post on the continued fraction convergents of pi. Motivated by Calvin Lin's comment whether a similar pattern exists for other constants, I checked $\sqrt{2}$. Its convergents ...
10
votes
1answer
411 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
5
votes
1answer
49 views

Convergence of a sequence to a value and not another.

I was tinkering with the following equation and produced an infinite nested fraction: $$ (x-6)(x-3)=0 $$ $$ x^2-9x+18=0 $$ $$ x=9-\frac{18}{x} $$ $$ x=9-\frac{18}{9-\frac{18}{9-\frac{18}{...}}} $$ ...
14
votes
2answers
287 views

Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$: $$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$ Any clues?
0
votes
3answers
150 views

Continued fractions proof?

Let $b_1=1$ and $$b_n=1+\frac{1}{1+b_{n-1}}$$ for $n\ge 2$. Note that $b_n \ge 1$ for all $n$ in $\mathbb N$. ($\mathbb N$ represents the positive integers) Show that $b_{2k-1}^2<2$ for all $k ...
7
votes
1answer
238 views

What causes the convergence of Iterated continued fractions from convergents?

Here is a small discovery I stumbled across a few weeks ago. I hope at least one person will find it interesting enough to help me. The iterated continued fractions from convergents (or convergents ...
10
votes
3answers
823 views

Proving the continued fraction representation of $\sqrt{2}$

There's a question in Spivak's Calculus (I don't happen to have the question number in front of me in the 2nd Edition, it's Chapter 21, Problem 7) that develops the concept of continued fraction, ...
5
votes
1answer
449 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.
10
votes
2answers
886 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 ...