Mixing
Non convergent simple continued fractions?
$begingroup$
Let $<a_0;a_1,a_2,dots>$ be an infinite sequence of integers such that
$$0<nimplies a_n>0.$$
For any natural $n$ we know there exists a convergent: one rational number $r_n$ such that it is equal to the simple continued fraction
$$r_n=a_0+frac{1}{a_1 +frac{1}{a_2 +frac{1}{ddots_{a_n+0}}}}$$
$r_n$ is the number given by the finite continued fraction of the original sequence truncated at $n$, $<a_0;a_1,a_2,dotsm,a_n>$.
The question is: given any sequence of positive integers, does the simple continued fraction always converge to a real number?
continued-fractions
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
$endgroup$
add a comment |
$begingroup$
Let $<a_0;a_1,a_2,dots>$ be an infinite sequence of integers such that
$$0<nimplies a_n>0.$$
For any natural $n$ we know there exists a convergent: one rational number $r_n$ such that it is equal to the simple continued fraction
$$r_n=a_0+frac{1}{a_1 +frac{1}{a_2 +frac{1}{ddots_{a_n+0}}}}$$
$r_n$ is the number given by the finite continued fraction of the original sequence truncated at $n$, $<a_0;a_1,a_2,dotsm,a_n>$.
The question is: given any sequence of positive integers, does the simple continued fraction always converge to a real number?
continued-fractions
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
$endgroup$
add a comment |
$begingroup$
Let $<a_0;a_1,a_2,dots>$ be an infinite sequence of integers such that
$$0<nimplies a_n>0.$$
For any natural $n$ we know there exists a convergent: one rational number $r_n$ such that it is equal to the simple continued fraction
$$r_n=a_0+frac{1}{a_1 +frac{1}{a_2 +frac{1}{ddots_{a_n+0}}}}$$
$r_n$ is the number given by the finite continued fraction of the original sequence truncated at $n$, $<a_0;a_1,a_2,dotsm,a_n>$.
The question is: given any sequence of positive integers, does the simple continued fraction always converge to a real number?
continued-fractions
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
$endgroup$
Let $<a_0;a_1,a_2,dots>$ be an infinite sequence of integers such that
$$0<nimplies a_n>0.$$
For any natural $n$ we know there exists a convergent: one rational number $r_n$ such that it is equal to the simple continued fraction
$$r_n=a_0+frac{1}{a_1 +frac{1}{a_2 +frac{1}{ddots_{a_n+0}}}}$$
$r_n$ is the number given by the finite continued fraction of the original sequence truncated at $n$, $<a_0;a_1,a_2,dotsm,a_n>$.
The question is: given any sequence of positive integers, does the simple continued fraction always converge to a real number?
continued-fractions
continued-fractions
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
asked Jan 22 at 19:51
Dr PotatoDr Potato
497
497
add a comment |
add a comment |
1 Answer
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$begingroup$
I just found the Continued Fraction Limit.
The answer is: yes!
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
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$begingroup$
I just found the Continued Fraction Limit.
The answer is: yes!
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
$endgroup$
add a comment |
$begingroup$
I just found the Continued Fraction Limit.
The answer is: yes!
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
$endgroup$
add a comment |
$begingroup$
I just found the Continued Fraction Limit.
The answer is: yes!
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
$endgroup$
I just found the Continued Fraction Limit.
The answer is: yes!
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
answered Jan 22 at 19:55
Dr PotatoDr Potato
497
497
add a comment |
add a comment |
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