P.S.
While Maxima makes it to the finish-line:
(%i2) sum(z^k / k, k, 1, inf);
inf
==== k
\ z
(%o2) > --
/ k
====
k = 1
(%i3) sum(z^k / k, k, 1, inf), simplify_sum;
(%o3) - log(1 - z)
it does so by cheating, giving that answer without requiring any assumption
that abs(z) < 1. Sympy does a better job at returning an answer (Piecewise)
that indicates that z's size matters to convergence.
On Tuesday, August 18, 2020 at 6:43:57 AM UTC-7, first last wrote:
>
> I'll try to clarify. Putting software aside momentarily, in pure math, for
> z real or complex with abs(z) < 1, for k from 1 to infinity, the following
> power-series summations hold:
>
> -log(1-x) = sum (z^k / k)
> log(1-x) = sum(-1 * z^k / k)
> log(1+x) = sum(-1 * z^k * (-1)^k / k)
>
> Maple and Mathematica can both do those, using their sum functions. (I'm
> not 100% confident in their handling of edge cases like z=-1 and z=+1, but
> my testing their has been haphazard.)
>
> I keep changing my story about Maxima, as I learn more about it. Yesterday
> I said Maxima cannot do those sums. Last night I learned Maxima can do
> them, via its "simplify_sum" feature, whose documentation is hidden in an
> obscure Chapter 84. Maxima manual's "Summation" chapter includes no mention
> of "simplify_sum". (I'm curious how many decades this Maxima
> documentation-bug has persisted without anyone simply moving "simplify_sum"
> to the chapter on sums. All 55 years of Maxima's history?)
>
> Sympy can sum those into Piecewise expressions:
>
> >>> sympy.summation(sympy.S('z^k / k'), sympy.S('(k, 1, oo)'))
> Piecewise((-log(1 - z), (z >= -1) & (z < 1)), (Sum(z**k/k, (k, 1, oo)),
> True))
> >>>
>
> The catch is, there's no way to ask Sympy "what's that Piecewise
> expression assuming abs(z) < 1"? Neither old nor new Sympy assumptions can
> express that query.
>
> I see two issues: Lack of functionality and room for
> documentation-improvement. I haven't designed and implemented my own
> assumptions-system, so I can't speak to its difficulty. Maxima's been
> worked on by countless geniuses for 55 years and still has an admittedly
> weak assumptions system; so maybe assumptions are especially hard for
> CAS-designers.
>
> The second issue is documentation. I have spent many months in many CAS's
> trying to sum the power series of log(1+x). I could have accomplished the
> same in a day is the CAS's had been documented better.
>
>
> On Monday, August 17, 2020 at 11:00:47 PM UTC-7 [email protected]
> wrote:
>
>> David,
>>
>> I'm not on this project, but I think it would save the devs time if you
>> would specify the sum you are referring to.
>>
>> -- Kind Regards,
>> Christian
>>
>> On Mon, Aug 17, 2020, 3:32 PM first last <[email protected]> wrote:
>>
>>> P.S. Maxima and Axiom are also unable to do this sum. Mathematica and
>>> Maple are able to do it.
>>>
>>> On Monday, August 17, 2020 at 3:30:26 PM UTC-7, first last wrote:
>>>>
>>>> I'll take the response as, there's no way to get Sympy to do this sum.
>>>>
>>>> On Friday, August 14, 2020 at 4:17:20 PM UTC-7, David Bailey wrote:
>>>>>
>>>>> Dear group,
>>>>>
>>>>> Am I correct that the write-up about assumptions found here relates to
>>>>> the old-style assumptions:
>>>>>
>>>>> https://docs.sympy.org/latest/modules/assumptions/assume.html
>>>>>
>>>>> Is there any documentation relating to the new assumptions?
>>>>>
>>>>> It would be really helpful if the documentation for old or new
>>>>> assumptions indicated which type it related to.
>>>>>
>>>>> David
>>>>>
>>>> --
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