A bit of exploration in the respective interpreters show that neither
Maxima, Sage, Giac, Fricas, Sympy nor Mathematica can give an expression
for this sum. Wolfram Alpha gives the numerical answer the OP reported, but
I have been unable to get any explanation about the process used ; graphics
appended to Wolfra Alpha's answer suggest that this numerical estimate is
the numerical value of the sum for a "sufficiently large" value of k, with
no indication about the choice of this "sufficiently large"...
As suggested by Nils Bruin, further work on numerical computation of sums
is granted...
HTH,
Le mardi 10 décembre 2024 à 17:34:40 UTC+1, Nils Bruin a écrit :
> Investigating the error shows that sympy didn't do anything with this sum:
>
> 671 try:
> --> 672 return result._sage_()
> 673 except AttributeError:
> 674 raise AttributeError("Unable to convert SymPy result
> (={}) into"
>
> ipdb> p result
> Sum(log(1 - 1/2**k), (k, 1, oo))
>
> so while solving the "NotImplemented" error would be nice, it wouldn't get
> you anything you didn't have already. The question then really is how to
> *numerically* evaluate an infinite sum in sage. As shown above, one way
> could be to use sympy for this. It may be that numerical sum evaluation
> hasn't received much attention in sage. The link below also suggests using
> sympy:
>
>
> https://ask.sagemath.org/question/32122/how-to-evaluate-the-infinite-sum-of-12n-1-over-all-positive-integers/
>
> I'm not sure how rigorous these methods are. You may have to do some
> analysis yourself to guarantee an error bound.
>
> On Tuesday, 10 December 2024 at 06:15:33 UTC-8 krts...@googlemail.com
> wrote:
>
>> Thanks for investigating. I tried to use sympy this way:
>>
>> sum(log(1-2^-k),k,1,oo,algorithm="sympy")
>>
>> but this failed for other reasons (NotImplementedError: conversion to
>> SageMath is not implemented)
>>
>>
>>
>> wdjo...@gmail.com schrieb am Dienstag, 10. Dezember 2024 um 14:58:49
>> UTC+1:
>>
>>> On Tue, Dec 10, 2024 at 8:43 AM 'OHappyDay' via sage-support <
>>> sage-s...@googlegroups.com> wrote:
>>>
>>>> I tried to evaluate the infinite product:
>>>>
>>>> prod((2^n-1)/2^n) (n=1,oo)
>>>>
>>>> by converting the product to a sum via logarithm:
>>>>
>>>> sum(log(1-2^-k),k,1,oo)
>>>>
>>>> The sum (and thus the product) should, according to WolframAlpha,
>>>> converge with a final value of about
>>>>
>>>> -1.24206
>>>>
>>>
>>> Sympy does it:
>>>
>>> sage: *from* *sympy* *import* oo, Sum, log, Product
>>>
>>> sage: p0 = Product((1-1/2^n), (n, 1, oo))
>>>
>>> sage: p0.evalf()
>>>
>>> 0.288788095086602
>>>
>>> sage: s0 = Sum( log(1-1/2^n), (n, 1, oo))
>>>
>>> sage: s0.evalf()
>>>
>>> -1.24206209481242
>>>
>>> sage: exp(-1.242) ## check
>>> 0.288806027885956
>>>
>>>
>>>>
>>>> This video (https://www.youtube.com/watch?v=KDyHJlNkov8) indicates
>>>> that the product converges with an irrational value.
>>>>
>>>> Sage reports that the sum is divergent.
>>>>
>>>> Ideas? Is this again a failure in Maxima?
>>>>
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>>>>
>>>
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