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1answer
59 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 ...
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1answer
27 views

How to write continued fraction as polynomial?

I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write $r(x)$ such that nominator and ...
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1answer
59 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
47 views

Is it possible to define a zero-set of $X$ to be the zero-set of some $f\in C^{*}(X)$?

It is possible to define a cozero-set of $X$ to be the cozero-set of some $f\in C^{*}(X)$, in fact; Every cozero-set in $X$ is the cozero-set of a function taking values in $[0, 1]$. $proof$: ...
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3answers
145 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 ...