Sum Divergent Series, I

by mnoonan

Every few months, I get in a particular mood that inspires me to look for reasonable finite values to assign to superficially divergent sums. I’d like to share some of them with you and start a discussion of just what “reasonable” means in this context. Finally, I have an open (and open-ended) question on very divergent series for you all to have a crack at.

I’m wrapping up an introductory calculus class at the moment, and in the course of doing other things I had the occasion to use the famous formula

$1 + x + x^2 + x^3 + \dots = \frac{1}{1 - x}$

I would bet that when most of us first saw this formula, we quickly started plugging in large values of $x$ without regard to such refined principles as “radius of convergence”. Setting $x = 1$ gives the unsurprising equation

$1 + 1 + 1 + 1 + \dots = \frac{1}{1 - 1} = \infty$

but $x = 2$ yields the much more impressive-looking

$1 + 2 + 4 + 8 + 16 + 32 + \dots = -1$

So my question to you, gentle reader: is this a cute little bit of nonsense, or an honest mathematical truth?

Let us get the most obvious complaint out of the way first. “How could we add only positive numbers and end up with a negative number? After all, we can prove that the sum of two positive numbers is positive!” Certainly this is true: any finite sum of positive numbers is still positive. But the axioms governing the arithmetic of real numbers involve only operations with a finite number of inputs, and the logical derivations we use to reason with these axioms all have a finite number of steps. So in fact, the axioms of the real numbers don’t tell us anything at all about what the sign of an expression such as $1 + 2 + 4 + 8 + 16 + \dots$ should be. We are liberated!

Well, not quite. Let us quickly recall how the closed expression for the sum of the geometric series is usually derived:

$(1 - x) \cdot (1 + x + x^2 + \dots)$

$= (1 + x + x^2 + \dots) - x \cdot (1 + x + x^2 + \dots) = 1$

Now, why would this proof hold for $x = 1/2$ but not $x = 2$ ? The canonical answer is that the series diverges when $|x| \geq 1$ , rendering the proof nonsense. More precisely, the set

$\{1, 1 + 2, 1 +2 + 4, 1 + 2 +4 + 8, 1 + 2 + 4 + 8 + 16, \dots \}$

has no limit points in the real numbers, so the partial sums are not approaching any real number. We should feel ashamed for blithely trying to do arithmetic with such a foul expression!

Then again, why should we listen to topological complaints about an algebraic statement with an algebraic proof? In fact, if instead of using the absolute value for our metric we used the 2-adic valuation, the sum honestly converges to $-1$ . We (and by “we” I mean “I”) might be inclined to think that the statement

$1 + x + x^2 + x^3 + \dots = \frac{1}{1 - x}$

is just true, though some poor benighted axiom systems (such as the real numbers) may be too unrefined to prove it for all values of $x$ . But forget about the abstraction for a moment: even your very-much-corporeal computer agrees that the sum of binary numbers

$1 + 10 + 100 + 1000 + \dots = -1$

The moral, maybe, is that we shouldn’t let intuition developed from the topology of the real number line get in the way of giving reasonable answers to infinite sums. Tomorrow, I’ll post more on methods for summing divergent series and what I might mean by “reasonable answers”. Also tomorrow: some potentially unreasonable answers for divergent sums, and a sum that seems to actually diverge.

This entry was posted on July 28, 2007 at 11:59 pm and is filed under Basic Grad Student, Matt, Undergraduate. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

26 Responses to “Sum Divergent Series, I”

Nugae Says:
July 29, 2007 at 4:24 am | Reply
I’m interested to watch how this continues. I got Hardy’s “Divergent Series” as a prize at school but never got much beyond the second chapter. H. points out that it is a Cheerful Fact that many of the “natural” ways of assigning sums of divergent series seem (when they work) to give the same answers, which suggests that there is something “out there” to be discovered; but that on the other hand the theory of divergent series only really started to progress once we had got past the idea of “what is the sum of this series?” to “what, reasonably, can we define the sum of this series to be”?

Hardy’s style is dry by modern standards but it has a depth of understanding behind it.
Josh Says:
July 29, 2007 at 3:41 pm | Reply
The sum of a geometric series is indeed an interesting place to start this discussion. For one thing, it brings up the “radius of convergence” issue for power series in general. After all, yes 1/(1-x) has an honest-to-goodness explosion to infinity at x=1, but it makes perfectly good sense at x=-1, and tells us (what my calc students certainly wanted to believe) that 1-1+1-1+1-1+… = 1/2. Plus, it’s interesting that you can get a geometric series, following this rule, to sum up to anything except 0. Could it be that 1+infty+infty^2+infty^3+… should equal 0?
Isabel Says:
July 29, 2007 at 6:25 pm | Reply
Another one that comes to mind (and maybe this is coming in the followup post):

zeta(s) is defined as 1^(-s) + 2^(-s) + 3^(-s) + … where that series converges.

zeta(-2) = zeta(-4) = … = 0.

So, letting s = -2, we get 1 + 4 + 9 + 16 + 25 + … = 0. What does this mean?
Tom Leinster Says:
July 29, 2007 at 7:30 pm | Reply
This question seems to be coming back into fashion! We had some discussion about it over at the n-Category Cafe:

http://golem.ph.utexas.edu/category/2007/07/return_of_the_euler_characteri.html

I’m afraid I caused this discussion to turn incoherent by asking some ill-posed questions; as Nugae says, there is apparently “something ‘out there’ to be discovered”, but it’s hard to put your finger on exactly what.

One powerful method that we talked about was this. Suppose you want to sum a divergent series

a_0 + a_1 + a_2 + … .

To attempt to do this, you can consider the power series

a_0 + a_1 x + a_2 x^2 + … .

If you’re lucky, this converges in a neighbourhood of 0. If you’re even luckier, it can be analytically continued to 1. If you’re luckier still, all such analytic continuations take the same value at 1 – which you can then declare to be the sum of the series.

This method gives you the expected answers for the geometric series talked about earlier in this thread. It doesn’t do the zeta values, though; e.g. it doesn’t tell you that 1 + 2 + 3 + … = -1/12.
Top Posts « WordPress.com Says:
July 29, 2007 at 8:02 pm | Reply
[…] Sum Divergent Series, I Every few months, I get in a particular mood that inspires me to look for reasonable finite values to assign to […] […]
mnoonan Says:
July 30, 2007 at 8:24 am | Reply
Thanks Tom, I read that post early on but missed out on all the great discussion that came out of it! As both you and Isabel have intuited, the next thing to look at is regularization schemes for doing these finite sums. But even in this case, I’m not totally convinced that we have the whole story — after all, if introducing topology (via limits of partial sums) causes problems for algebraic divergent sums*, I’m skeptical of placing analytic methods on too high a pedestal. On the other hand, Abel and zeta regularization really do work incredibly well on these sums. So are the analytic methods and end to themselves, or do they just happen to sometimes compute the morally correct* sum?

Nugae: thanks for the pointer to the book. I had forgotten about it! It turns out that the full text is available to everybody on earth through the Internet Archive. I’ll add a link in the next post.

Josh: I’m trying to sum a nasty divergent series as we speak, and it looks like the equation 1 + infty + infty^2 + … = 0 might take care of enough divergences to yield a reasonable sum. So it might just be true*!

* whatever that means
Adding, Multiplying and the Mellin Transform « The Everything Seminar Says:
August 3, 2007 at 3:18 am | Reply
[…] Multiplying and the Mellin Transform Matt’s entertaining posts on divergent series have inspired me to contribute my own two cents. In his posts, the central […]
Anonymous Says:
September 16, 2007 at 4:04 pm | Reply
The sum of the powers of two reminds me of this. hakmem entry.
Jim Says:
September 16, 2007 at 4:05 pm | Reply
The sum of the powers of two reminds me of this

http://www.inwap.com/pdp10/hbaker/hakmem/hacks.html#item154

hakmem entry.
Lubos and divergent series « The Gauge Connection Says:
February 23, 2009 at 6:42 am | Reply
[…] https://cornellmath.wordpress.com/2007/07/28/sum-divergent-series-i/ […]
Yasiru Says:
February 27, 2009 at 9:25 am | Reply
See my blog at,
http://mathrants.blogspot.com
Daichi Handa (@dpendh) Says:
December 9, 2011 at 11:45 am | Reply
filmy za free Says:
February 5, 2012 at 12:04 pm | Reply
filmy za free…

[…]Sum Divergent Series, I « The Everything Seminar[…]…
here Says:
April 9, 2012 at 3:21 am | Reply
here…

[…]Sum Divergent Series, I « The Everything Seminar[…]…
www.e-Boekhouder.be Says:
April 19, 2013 at 9:27 am | Reply
Does your site have a contact page? I’m having problems locating it but, I’d
like to shoot you an email. I’ve got some ideas for your blog you might be interested in hearing. Either way, great site and I look forward to seeing it develop over time.
vip.zzia.edu.cn Says:
April 27, 2013 at 5:37 pm | Reply
Hi mates, its great article on the topic of educationand
completely explained, keep it up all the time.
seo Says:
April 30, 2013 at 5:44 am | Reply
Howdy would you mind letting me know which webhost you’re working with? I’ve
loaded your blog in 3 completely different browsers and I must say this
blog loads a lot faster then most. Can you recommend
a good internet hosting provider at a fair price?
Thanks, I appreciate it!
Filmasy24.pl - Filmy za darmo Says:
June 25, 2013 at 12:59 am | Reply
Asking questions are really pleasant thing if you are not understanding something totally,
however this article offers nice understanding yet.
Augustina Says:
July 8, 2013 at 11:40 am | Reply
hello there and thank you for your info – I’ve definitely picked up anything new from right here. I did however expertise several technical issues using this web site, since I experienced to reload the site lots of times previous to I could get it to load correctly. I had been wondering if your web hosting is OK? Not that I am complaining, but slow loading instances times will very frequently affect your placement in google and could damage your high-quality score if advertising and marketing with Adwords. Anyway I’m
adding this RSS to my e-mail and could look out for a lot more of your respective exciting content.
Make sure you update this again soon.
adhocspace.Com Says:
July 19, 2013 at 7:30 am | Reply
First of all I would like to say superb blog!
I had a quick question in which I’d like to ask if you don’t mind.
I was curious to find out how you center yourself and clear
your thoughts before writing. I’ve had a hard time clearing my thoughts in getting my thoughts out. I truly do take pleasure in writing however it just seems like the first 10 to 15 minutes are usually lost simply just trying to figure out how to begin. Any suggestions or hints? Many thanks!
corporate gifts Says:
July 19, 2013 at 11:45 am | Reply
hey there and thank you for your info – I
have definitely picked up anything new from right here.

I did however expertise several technical issues using this
site, as I experienced to reload the site lots of times
previous to I could get it to load properly.
I had been wondering if your web host is OK? Not that I’m complaining, but sluggish loading instances times will sometimes affect your placement in google and can damage your high-quality score if ads and marketing with Adwords. Well I’m adding this RSS to
my email and could look out for a lot more of your respective interesting content.
Ensure that you update this again very soon.
office rental Says:
July 29, 2013 at 10:31 pm | Reply
Thanks for any other wonderful article. The place else may
anybody get that kind of information in such a
perfect manner of writing? I have a presentation subsequent week, and I am at the search for such information.
On -1/12, adding infinitely many numbers, and Phil Plait’s rash and incorrect claims | konstantinkakaes.com Says:
January 18, 2014 at 12:54 am | Reply
[…] now this does not “equal” 1/(1-2)=1/(-1)=-1 in any sense other than by some sort of loose analogy, as this lucid, thoughtful, and mildly technical blog post by Matt Noonan explains. […]
Синиша Бубоња Says:
February 14, 2015 at 5:07 pm | Reply
Dear Mnoonan,
First of all sorry for my bad English. Your three posts about divergent series is excellent! I found a way to sum divergent series and determine the limits of function in their singular points. I discovered that the method can be applied to compute divergent integrals.

https://m4t3m4t1k4.wordpress.com/2015/02/14/general-method-for-summing-divergent-series-determination-of-limits-of-divergent-sequences-and-functions-in-singular-points-v2/

Sincerely, Sinisa
seo uzmanı Says:
November 1, 2016 at 7:32 am | Reply
Uzun süredir bulamadığım içerikdi teşekkürlerimi sunarım
piece of shit Says:
March 26, 2021 at 12:49 pm | Reply
disgusting mother fucker