# Tagged Questions

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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### Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\large\frac { 1 }{ 1+\frac { 1 }{ 2+\frac { 1 }{ 3+\frac { 1 }{ 4+\dots } } } }$$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I ...
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### Writing continued fractions of irrational numbers as infinite series

Infinite sums have been formulated for famous irrational numbers, such as $\pi, \phi,e,\sqrt2$ and a few others that can be listed here and here: Here are some examples: (There are more examples ...
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### Periodicity of the continued fraction of a square root

Writing $\sqrt{n}=[a_0; a_1, a_2, \dots ]$, at which $a_i$ does the period start? Is it $a_1$? I just put "for some $n\ge 1$, where $a_{n-1}=a_i$", is that a good enough answer?
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### Can I make an infinte sum using rational numbers that makes an irrational but not transcendental number?

I looked a lot on the internet for examples and I tried to do it myself, but I haven't seen any infinite sums of rational numbers that equal for example something like square root of 10 or cube root ...
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### How to use Euler's continued fraction formula?

I am trying to convert some continued fractions to series by using Euler's continued fraction formula (see the link to Wikipedia). But there is something I obviously misunderstood in it. Take for ...
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### Continued Fractions : Under which branch of mathematics do they come?

I wanted to know in which branch of Mathematics do Continued Fraction come? By branch I mean for example Geometry or Differential Equation are a branch of maths so is there any particular branch of ...
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### conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
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### Good book for self study of Continued Fractions

Does anyone have a recommendation for a rigorous while readable book to use for the self study of continued fractions? PS - As examples of "rigorous while readable book" for self-learning, A. ...
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### Negative solution for a positive continued fraction

$$x=1+\cfrac{1}{1+\cfrac{1}{1+...}}\implies x=1+\frac{1}{x}\implies x=\frac{1\pm \sqrt{5}}{2}$$ Can the negative solution be considered as a solution? If yes, how is it possible to have a negative ...
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### Newton's method for square roots 'jumps' through the continued fraction convergents

I know that Newton's method approximately doubles the number of the correct digits on each step, but I noticed that it also doubles the number of terms in the continued fraction, at least for square ...
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### How to make continued fractions of any number?

I recently found an continued fraction representation of $\pi$, and I wondered how can I make an continued fraction that converges into a number? The MAIN question is: how do you make a continued ...
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### Does the continued fraction for $e^{3/n}$ have a pattern?

Wikipedia has patterns for the simple continued fractions $e^{1/n},e^{2/n}$, which made me wonder whether there is one known for $e^{3/n}?$ (by pattern, I mean that the partial quotients $a_n$ can ...
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