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
I would bet that when most of us first saw this formula, we quickly started plugging in large values of without regard to such refined principles as “radius of convergence”. Setting gives the unsurprising equation
but yields the much more impressive-looking
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 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:
Now, why would this proof hold for but not ? The canonical answer is that the series diverges when , rendering the proof nonsense. More precisely, the set
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 . We (and by “we” I mean “I”) might be inclined to think that the statement
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 . But forget about the abstraction for a moment: even your very-much-corporeal computer agrees that the sum of binary numbers
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.