Talk:Simple Infinite Continued Fraction Converges

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Rename

A better name is Simple Infinite Continued Fraction Converges (which is what is proven here). The value is by definition the limit of convergents. --barto (talk) 16:20, 17 July 2017 (EDT)

Done. --barto (talk) (contribs) 14:12, 18 January 2018 (EST)

Proof for Cauchy sequence

Concerning the question about $\langle C_n\rangle$ being a Cauchy sequence, this should follow from $\dfrac 1 {k^2}$ being summable: we have $:\displaystyle \left|C_n - C_m\right| \le \left|C_n - C_{n+1}\right| + \ldots + \left|C_{m-1} - C_m\right| \le \dfrac 1 {n^2} + \ldots + \dfrac 1 {\left(m-1\right)^2} \le \sum_{k \mathop = n}^\infty \dfrac 1 {k^2} \rightarrow 0$ as $n \rightarrow \infty$ since the sum is the tail of a convergent series. --Trenta3

Indeed (and welcome!). Feel free to add it to the proof. (It also follows from the stronger fact that $q_n\geq F_n$, the $n$th Fibonacci number.) --barto (talk) 07:05, 4 August 2017 (EDT)