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Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg132251] Re: How to show 1+2+3+ ... = -1/12 using Mathematica's symbols?
*From*: Andrzej Kozlowski <akozlowski at gmail.com>
*Date*: Thu, 23 Jan 2014 03:35:05 -0500 (EST)
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*References*: <20140119075641.AB76D69EE@smc.vnet.net> <20140120090105.33E5969EA@smc.vnet.net> <20140121080228.9BF4B69D9@smc.vnet.net> <8E65D7C9-8567-4032-B629-46FD621486DF@math.umass.edu> <0CA1B3F8-D617-407E-9027-A13821882AE0@gmail.com> <93BFFEFC-5DE0-4BF2-BAAC-2A4BB6A8C666@math.umass.edu> <20140122083126.3EFDC69D9@smc.vnet.net> <A6BE0E9D-33CE-42FF-B842-A6B3478D4011@math.umass.edu>
You are quite right. I was thinking of a different book by Knopp Infinite Sequences and Series. The one that you were referring to is more advanced and is called Theory and Applications of Infinite Series and it does indeed deal with Divergent Series.
As for Cartwight's theorems, as far as i can tell from a quick glance (I read Hardy's book quite a long time ago and not in detail) the section of the book where they are proved is concerned with the relationship between Abel summations (A,n^p) and (A,n^q) where q<p. (The notation (A,n^p) means that we consider Abel summation given by the sequence 0< 1^p < 2^p < 3^p < =85 n^p ). The theorems show that in such cases the method (A,n^q) is weaker than (A,n^p) in the sense that it is less likely to be applicable but when it is applicable it is more effective.
The precise statements are too technical for posting here.
(I might have misunderstood something, but roughly this is what the theorems seem to be about).
Andrzej
On 22 Jan 2014, at 16:55, Murray Eisenberg <murray at math.umass.edu> wrote:
> I just checked a freely-available on-line scan of an English edition of Knopp, and he definitely discusses summation methods for divergent series, in Chapter XIII. In fact, when I originally consulted the German edition, it was to learn about Cesaro summability of divergent series.
>
> (Unfortunately, Hardy's Divergent Series still seems to be under copyright =97 at least that's the case with a 1992 American Mathematical Society reissue of a Chelsea Publications 2nd edition. And copies of the 1st edition are just as expensive.)
>
> [The table of contents of Hardy's book lists an appendix Two theorems of M.L. Cartwright. Does anbody reading this post happen to know what they concern? (I'm familiar with some of Dame Cartwright's work on differential equations and dynamical systems and even heard her lecture about that some 50 years ago.)]
>
>
> On Jan 22, 2014, at 3:31 AM, Andrzej Kozlowski <akozlowski at gmail.com> =
wrote:
>
>> I think Knopp's book only discusses summation of convergent series =
(in the ordinary sense). The only book I know that gives a full and =
rigorous treatment of various methods of summation of divergent series =
is the book by Hardy Divergent Series that I have already mentioned. =
Like all of Hardy's books it is very elegantly and readably written (of =
course as long as you are one of those people who think that a whole =
page filled only with mathematical formulas can be called Creadable ;-) =
).
>> In fact I only have a Russian translation of this book from 1951 (the =
original was published in 1949) and the translator states that no other =
comprehensive text on this topic exists. I think this is still the case =
more than half a century later.
>>
>> The most classic book in English on the Cconventional theory of =
infinite series is probably Bromwich An Introduction to the Theory of =
Infinite Series, an electronic version of which can be downloaded freely =
and legally from the Internet. Obviously it does not discuss Ramanujan's =
ummantion. In fact, this is the text mentioned in the often quoted =
letter from Ramanujan to Hardy:
>>
>> "Dear Sir, I am very much gratified on perusing your letter of the =
8th February 1913. I was expecting a reply from you similar to the one =
which a Mathematics Professor at London wrote asking me to study =
carefully Bromwich's Infinite Series and not fall into the pitfalls of =
divergent series. ... I told him that the sum of an infinite number of =
terms of the series: 1 + 2 + 3 + 4 + ,,, -1/12 under my theory. If I
>> tell you this you will at once point out to me the lunatic asylum as =
my goal.
>>
>> ;-)
>>
>> Andrzej
>>
>>
>> On 21 Jan 2014, at 23:57, Murray Eisenberg <murray at math.umass.edu> =
wrote:
>>
>>> Adrzej,
>>>
>>> Do I correctly recall that Ramanujan summation is _not_ discussed in =
Knopp's classic book on infinite series?
>>>
>>> (I think I may have last closely studied it the German war-time =
recycled-paper edition Theorie ind Anwendung der unendlichen Reihen in =
1960, for a seminar with Abram Besicovitch and I.J. Schoenberg.)
>>>
>>> Murray
>>>
>>> On Jan 21, 2014, at 3:24 PM, Andrzej Kozlowski =
<akozlowski at gmail.com> wrote:
>>>
>>>>
>>>>
>>>>
>>>> On 21 Jan 2014, at 19:58, Murray Eisenberg <murray at math.umass.edu> =
wrote:
>>>>
>>>>> Andrzej,
>>>>>
>>>>> Drat, I tried each documented value for the Regularization option =
except that one!
>>>>
>>>> Yes, the name CDirichlet for this summation (or =
=E2=80=9Cregularization=E2=80=9D) method seems to me =
non-standard but it =
>> was the only one that suggested a relation with the zeta function.
>>>> Hardy in =E2=80=9CDivergent Series=E2=80=9D called this =
summation =
>> method =E2=80=9CRamanujan summation=E2=80=9D, since =
Ramanujan used it =
>> all the time and obtained lots of formulas with it, although the =
>> classic =E2=80=9Cequality=E2=80=9D in this subject of =
this thread goes =
>> back to Euler.
>>>>
>>>> Andrzej
>>>>
>>>>
>>>>>
>>>>> On Jan 21, 2014, at 3:02 AM, Andrzej Kozlowski =
>> <akozlowski at gmail.com> wrote:
>>>>>
>>>>>> Note that:
>>>>>>
>>>>>> In[25]:= Sum[n, {n, 1, Infinity}, Regularization -> =
"Dirichlet"]
>>>>>>
>>>>>> Out[25]= -(1/12)
>>>>>>
>>>>>> This is of course, perfectly correct ;-)
>>>>>>
>>>>>> Andrzej
>>>>>>
>>>>>> On 20 Jan 2014, at 10:01, Murray Eisenberg =
<murray at math.umass.edu> =
>> wrote:
>>>>>>
>>>>>>> You may try the Regularization option for Sum, but it doesn't =
seem =
>> to give any finite result for that divergent series.
>>>>>>>
>>>>>>> On the other hand, the video to which you refer relies =
ultimately =
>> upon using Ces=E0ro-summability of 1 - 1 + 1 - 1 _ . . . , which =
you =
>> may implement in Mathematica as:
>>>>>>>
>>>>>>> Sum[(-1)^n, {n, 0, \[Infinity]}, Regularization -> =93Cesaro"]
>>>>>>> (* 1/2 *)
>>>>>>>
>>>>>>> [The video to which you refer is disingenuous in not saying =
>> up-front that it's not using ordinary summability but some other =
form(s) =
>> of summability. (The merest hint is a brief glimpse of a page of a =
text =
>> on String Theory where the formula
>>>>>>> 1 + 2 + 3 + . . . = -1/12 is displayed just below a line =
>> referring to renormalization.)
>>>>>>>
>>>>>>> As it stands, that video, in my mind, is deleterious to =
>> understanding of the mathematics of infinite series destructive of =
trust =
>> in mathematics: it manipulates divergent series as if they were =
>> convergent.]
>>>>>>>
>>>>>>>
>>>>>>> On Jan 19, 2014, at 2:56 AM, Matthias Bode <lvsaba at hotmail.com> =
=
>> wrote:
>>>>>>>
>>>>>>>>
>>>>>>>> Hola,
>>>>>>>>
>>>>>>>> I came across this video (supported by the Mathematical =
Sciences =
>> Research Institute* in Berkeley, California):
>>>>>>>>
>>>>>>>> http://www.numberphile.com/videos/analytical_continuation1.html
>>>>>>>>
>>>>>>>> Could the method shown in this video be replicated using =
>> Mathematica symbols such as Sum[] &c.?
>>>>>>>>
>>>>>>>> Best regards,
>>>>>>>>
>>>>>>>> MATTHIAS BODES 17.36398=B0, W 66.21816=B0,2'590 m. AMSL.
>>>>>>>>
>>>>>>>> *) http://www.msri.org/web/msri
>>>>>>>>
>>>>>>>
>>>>>>> Murray Eisenberg =
>> murray at math.umass.edu
>>>>>>> Mathematics & Statistics Dept.
>>>>>>> Lederle Graduate Research Tower phone 240 246-7240 (H)
>>>>>>> University of Massachusetts
>>>>>>> 710 North Pleasant Street
>>>>>>> Amherst, MA 01003-9305
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>
>>>>>>
>>>>>
>>>>> =E2=80=94=E2=80=94
>>>>> Murray Eisenberg =
>> murray at math.umass.edu
>>>>> Mathematics & Statistics Dept.
>>>>> Lederle Graduate Research Tower phone 240 246-7240 (H)
>>>>> University of Massachusetts
>>>>> 710 North Pleasant Street
>>>>> Amherst, MA 01003-9305
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>
>>>
>>> =E2=80=94=E2=80=94
>>> Murray Eisenberg =
murray at math.umass.edu
>>> Mathematics & Statistics Dept.
>>> Lederle Graduate Research Tower phone 240 246-7240 (H)
>>> University of Massachusetts
>>> 710 North Pleasant Street
>>> Amherst, MA 01003-9305
>>>
>>>
>>>
>>>
>>>
>>>
>>
>>
>
> =97=97
> Murray Eisenberg murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower phone 240 246-7240 (H)
> University of Massachusetts
> 710 North Pleasant Street
> Amherst, MA 01003-9305
>
>
>
>
>
>
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