1
vote
1answer
23 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
1
vote
1answer
45 views

Finite Continued Fraction Proof

Let $\alpha = \left[a_0, a_1, a_2,\cdots,a_n\right]$ be a finite continued fraction with $a_0 > 0$ and let $C_i = p_i/q_i$ be the convergent of $\alpha$. If $i\ge 1$, prove that ...
5
votes
1answer
49 views

Convergence of a sequence to a value and not another.

I was tinkering with the following equation and produced an infinite nested fraction: $$ (x-6)(x-3)=0 $$ $$ x^2-9x+18=0 $$ $$ x=9-\frac{18}{x} $$ $$ x=9-\frac{18}{9-\frac{18}{9-\frac{18}{...}}} $$ ...
9
votes
0answers
159 views

Continued fraction with prime reciprocal entries

We know that the reciprocals of the primes form a divergent series. We also know that a necessary and sufficient condition for a continued fraction to converge is that its entries diverge as a series. ...
2
votes
1answer
92 views

Why are continued fractions for irrational numbers always convergent?

Like in the title: Why are continued fractions for irrational numbers (i.e. infinite fractions) always convergent?
5
votes
1answer
976 views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
5
votes
2answers
499 views

Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?

I have been looking at concise ways to represent irrational numbers using only integers. I was thinking about base $\phi$ (golden ratio base) and how it can represent the quadratic extension of the ...