For questions on continued fractions.

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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 ...
16
votes
3answers
940 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 ...
5
votes
1answer
246 views

How to find the number of continued fraction from a periodic representation?

Problem Find the number that represented by $[2,2,2 \ldots]$ I know it wasn't difficult, but I was absent the last two classes. So I just want to make sure that I got it right. My attempt was, ...
3
votes
1answer
145 views

How to find continued fraction of the form $a\sqrt{b}$?

For the form $\sqrt{b}$, I could just apply the recursive quadratic formula: $$P_{k+1} = a_kQ_k - P_k$$ $$Q_{k+1} = \dfrac{d - P^2_{k+1}}{Q_k}$$ $$\alpha_k = \dfrac{P_k + \sqrt{d}}{Q_k}$$ ...
12
votes
1answer
471 views

Deriving a trivial continued fraction for the exponential

Lately, I learned about the following continued fraction for the exponential function: $$\exp(x)=1+\cfrac{x}{1-\cfrac{x/2}{1+x/2-\cfrac{x/3}{1+x/3-\cfrac{x/4}{1+x/4-\dots}}}}$$ I thought it was ...
6
votes
2answers
217 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 ...
10
votes
3answers
859 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, ...
2
votes
1answer
276 views

Solving equations using continued fractions?

We solve the pell equation using the continued fraction for square root of 2. What equations can we solve using the continued fraction of cube roots (and other numbers too)?
7
votes
0answers
455 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+ ...
6
votes
1answer
377 views

What is the length of a continued fraction expansion of a rational number?

I was reviewing quantum factorization and am slightly unclear on a classical detail of order-finding. Given a (suitably nice) periodic function $f$ with unknown period $r$ and a power of two $N > ...
10
votes
1answer
564 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 ...
5
votes
2answers
312 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
459 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.
8
votes
3answers
718 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
891 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 ...
15
votes
2answers
823 views

Continued fraction for $\frac{1}{e-2}$

A couple of years ago I found the following continued fraction for $\frac1{e-2}$: $$\frac{1}{e-2} = 1+\cfrac1{2 + \cfrac2{3 + \cfrac3{4 + \cfrac4{5 + \cfrac5{6 + \cfrac6{7 + \cfrac7{\cdots}}}}}}}$$ ...
5
votes
1answer
754 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 ...