Re: [math-fun] Egyptian Fractions (was Re: Messages in pi (cont'd))
There is the greedy-harmonic-series-with-signs method: Let C be the irrational real number to approximate; say C > 0. Then let p1 be the least integer for which 1 + 1/2 + 1/3 + ... + 1/p1 > C. Now let n1 be the least integer > p1 for which 1 + 1/2 + 1/3 + ... + 1/p1 - (1/(p1+1) + 1/(p1+2) + ... + 1/n1) < C Continue this way, alternately adding and subtracting groups of consecutive terms, resulting in a signed harmonic series that converges to C. —Dan Keith Lynch wrote: ----- ..... What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original. -----
This applies to any divergent series, right? And the fun thing is that any C has a series with any initial subsequence. I suppose this means any C has countably many sequences, although it’s not clear that taking a finite initial subsequence and then turning greedy covers all of them… - Cris
On Dec 10, 2018, at 3:43 PM, Dan Asimov <dasimov@earthlink.net> wrote:
There is the greedy-harmonic-series-with-signs method:
Let C be the irrational real number to approximate; say C > 0.
Then let p1 be the least integer for which
1 + 1/2 + 1/3 + ... + 1/p1 > C.
Now let n1 be the least integer > p1 for which
1 + 1/2 + 1/3 + ... + 1/p1 - (1/(p1+1) + 1/(p1+2) + ... + 1/n1) < C
Continue this way, alternately adding and subtracting groups of consecutive terms, resulting in a signed harmonic series that converges to C.
—Dan
Keith Lynch wrote: ----- ..... What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original. -----
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On Mon, Dec 10, 2018 at 6:21 PM Cris Moore <moore@santafe.edu> wrote:
This applies to any divergent series, right? And the fun thing is that any C has a series with any initial subsequence. I suppose this means any C has countably many sequences, although it’s not clear that taking a finite initial subsequence and then turning greedy covers all of them…
The eventually greedy ones do not cover all of them. Suppose I want to make a series that converges to X, but is not eventually greedy. All I have to do is, when deciding the sign of the next term, once in a while but infinitely often, make a choice that doesn't move the total-so-far towards X. For example, I could decide that when the next summand will make the partial sum cross over from < x to > x, make it negative instead. This will give a sequence that converges to x, but whose partial summands never exceed x. The decision as to whether to do this can be made independently each time the next summand will give a total > x and the previous summand was < x. So this tiny subset of the sequences that converge to x (the ones where the only time the next summand is not given a sign that will move closer to x is when the result would change the sign of (total-so-far - x)) are already uncountable, since the infinite number of binary choices can be encoded as a real number.
- Cris
On Dec 10, 2018, at 3:43 PM, Dan Asimov <dasimov@earthlink.net> wrote:
There is the greedy-harmonic-series-with-signs method:
Let C be the irrational real number to approximate; say C > 0.
Then let p1 be the least integer for which
1 + 1/2 + 1/3 + ... + 1/p1 > C.
Now let n1 be the least integer > p1 for which
1 + 1/2 + 1/3 + ... + 1/p1 - (1/(p1+1) + 1/(p1+2) + ... + 1/n1) < C
Continue this way, alternately adding and subtracting groups of consecutive terms, resulting in a signed harmonic series that converges to C.
—Dan
Keith Lynch wrote: ----- ..... What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original. -----
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Andy.Latto@pobox.com
I see - so for any real X there are countably many choices we can make, while compensating for these choices by being greedy in between. Very nice! C
On Dec 12, 2018, at 7:43 AM, Andy Latto <andy.latto@pobox.com> wrote:
On Mon, Dec 10, 2018 at 6:21 PM Cris Moore <moore@santafe.edu> wrote:
This applies to any divergent series, right? And the fun thing is that any C has a series with any initial subsequence. I suppose this means any C has countably many sequences, although it’s not clear that taking a finite initial subsequence and then turning greedy covers all of them…
The eventually greedy ones do not cover all of them. Suppose I want to make a series that converges to X, but is not eventually greedy. All I have to do is, when deciding the sign of the next term, once in a while but infinitely often, make a choice that doesn't move the total-so-far towards X. For example, I could decide that when the next summand will make the partial sum cross over from < x to > x, make it negative instead. This will give a sequence that converges to x, but whose partial summands never exceed x.
The decision as to whether to do this can be made independently each time the next summand will give a total > x and the previous summand was < x. So this tiny subset of the sequences that converge to x (the ones where the only time the next summand is not given a sign that will move closer to x is when the result would change the sign of (total-so-far - x)) are already uncountable, since the infinite number of binary choices can be encoded as a real number.
- Cris
On Dec 10, 2018, at 3:43 PM, Dan Asimov <dasimov@earthlink.net> wrote:
There is the greedy-harmonic-series-with-signs method:
Let C be the irrational real number to approximate; say C > 0.
Then let p1 be the least integer for which
1 + 1/2 + 1/3 + ... + 1/p1 > C.
Now let n1 be the least integer > p1 for which
1 + 1/2 + 1/3 + ... + 1/p1 - (1/(p1+1) + 1/(p1+2) + ... + 1/n1) < C
Continue this way, alternately adding and subtracting groups of consecutive terms, resulting in a signed harmonic series that converges to C.
—Dan
Keith Lynch wrote: ----- ..... What are some exotic ways to approximate real numbers to any desired precision? Extra points if you can come up with something completely original. -----
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Andy.Latto@pobox.com
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participants (3)
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Andy Latto -
Cris Moore -
Dan Asimov