# Questions tagged [continued-fractions]

A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

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### Infinite continued fraction for $\pi$ [closed]

Recently I have found the following infinite continued fraction for $\pi$ : $$\pi=4-\cfrac{2}{1+\cfrac{1}{1- \cfrac{1}{1+\cfrac{2}{ 1-\cfrac{2 }{ 1+\cfrac{3}{ 1-\cfrac{3 }{\cdots}}}}}}}$$ Where can I ...
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### The measure of a set of continued fractions. [duplicate]

What would be the measure of the set of continued fractions which only contain 1 or 2 in the denominator, including infinite fractions? If I understand measure correctly, the set of continued ...
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### Continued fractions convergents proofs

Given a continued fraction, let $\frac{p_n}{q_n}$ and $\frac{p_{n+1}}{q_{n+1}}$ be two consecutive convergents. The proof states: $|x - \frac{p_n}{q_n}| \leq \frac{2}{2q_n}$ Using the ...
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### Divergence of the generalized continued fraction $1+ \frac{-1\mid}{\mid1}+\frac{-1\mid}{\mid1}+\frac{-1\mid}{\mid1}+\dots$

I learned about continued fraction, and I wondered whether this form of fraction can be a complex number. $$1-\cfrac1{1-\cfrac1{1-\cfrac1{1-\cfrac1\ddots}}}$$ This cannot be convergent, but I can't ...
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### If $y$ is a quadratic irrational, then there exists $M > 0$ such that $\forall p, q$, $q > 0$, $\left| \frac{p}{q} - y \right| > \frac{M}{q^2}$.

Suppose $y$ is a quadratic irrational. I want to show that there exists $M>0$ such that for all $p, q$, $q>0$, $$\left| \frac{p}{q} - y \right| > \frac{M}{q^2}.$$ I'd like a reminder on how ...
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### Convert Rational Function to Finite Continuted Functions.

If we have a finite continued rational functions, it is easy to clear the demoninator and write as a rational function (quotient of polynomials). For example, \frac{1}{x+\frac{1}{x+\frac{1}{x+1}}}=\...
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