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 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.

26 Responses to “Sum Divergent Series, I”

  1. Nugae Says:

    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.

  2. Josh Says:

    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?

  3. Isabel Says:

    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?

  4. Tom Leinster Says:

    This question seems to be coming back into fashion! We had some discussion about it over at the n-Category Cafe:

    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.

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  6. mnoonan Says:

    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

  7. Adding, Multiplying and the Mellin Transform « The Everything Seminar Says:

    […] 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 […]

  8. Anonymous Says:

    The sum of the powers of two reminds me of this. hakmem entry.

  9. Jim Says:

    The sum of the powers of two reminds me of this

    hakmem entry.

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  23. On -1/12, adding infinitely many numbers, and Phil Plait’s rash and incorrect claims | Says:

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  24. Синиша Бубоња Says:

    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.

    Sincerely, Sinisa

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