# Tagged Questions

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### How to prove continued fraction convergents of a number

Let $x=1+\sqrt{3}$. Prove that in pairs the continued fraction convergents of $x$ are $a_n$/$b_n$ < x < $c_n$/$d_n$ where $a_1$ = 2, $b_1$ = 1, $c_1$ = 3, $d_1$ = 1, $a_{n+1}$ = 2$c_n$ + $a_n$, ...
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### Intermediate convergents of a continued fraction.

I have been studying continued fractions and convergent's properties, and i have a questions about "intermediate convergents" I have read that the expression of the intermediate convergents (those ...
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### About continued fractions as best rational approximations

I'm reading this notes about continued fractions: http://www.math.jacobs-university.de/timorin/PM/continued_fractions.pdf I had no problems understanding everything there, except one thing that has ...
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### Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$

I'm currently a high school junior enrolling in AP Calculus, I found this website that's full of "math geeks" and I hope you can give me some clues on how to solve this problem. I'm pretty desperate ...
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I have a question concerning continued fractions: If we have $\gamma \in \mathbb{R} \setminus \mathbb{Q}$ and $\gamma=\langle a_0;a_1,a_2,\dotsc\rangle$. Why do we get $$\frac1\gamma = \langle ... 3answers 417 views ### Continued fraction: Show \sqrt{n^2+2n}=[n; \overline{1,2n}] I have to show the following identity (n \in \mathbb{N}):$$\sqrt{n^2+2n}=[n; \overline{1,2n}] I had a look about the procedure for $\sqrt{n}$ on Wiki, but I don't know how to transform it to ...
I have to prove the following: Let $\alpha=[a_0;a_1,a_2,...,a_n]$ and $\alpha>0$, then $\dfrac1{\alpha}=[0;a_0,a_1,...,a_n]$ I started with ...
### Show $p(n)=n(p_{n-1}+p_{n-2})+(n-1)p_{n-3}+(n-2)p_{n-4}+…+3p_1+2p_0+2$
I have to show the following: Let $N_k=\frac{p_k}{q_k}$ with $\alpha=\langle 1;2,3,4,...,n,n+1\rangle$ and $n \in \mathbb{N}$. Then $\forall n \in \mathbb{N}$ with $n\geq 3$, ...