Matt’s Sunday Function this week is a weird one, a series that is only conditionally convergent:
So the sum of the infinite series, by inexorable logic, is both ln(2) and ln(2)/2. How is this possible?
Of course it isn’t. The flaw in our logic is the assumption that the series has a definite sum – in the mathematical parlance, that it’s absolutely convergent. This series is not, it’s only conditionally convergent. In fact you can show (the great G.F.B. Riemann was the first) that with judicious rearrangement, you can get this series to converge to anything at all. As such it’s only meaningful to talk about the sum of this series if you specify the particular ordering you happen to be working with. For finite N the ordering doesn’t matter so long as you include the same terms, but you can’t do the calculus to find the infinite-N limit without a specific ordering.
It’s unusual to encounter this sort of series in physics. Most of our series are either absolutely convergent or simply divergent by any standard. But math is weird, and you can’t always assume that things work the way you intuitively expect. You have to rigorously check your assumptions.
This is one of the big differences between physics and math, and why it’s a little tricky for me to teach very formal mathematical classes– I’m very much a swashbuckling experimentalist, used to plunging ahead secure in the knowledge that an actual measurement will give a definite result of some sort, and not worrying about fussy details of the formalism.
Matt’s last paragraph reminds me of a post-doc I knew in grad school, who once helped his sister with math homework on series expansions. He asked her later how she had done on the assignment, and she said “Terrible. The prof said I did it like a physicist.”
In math, you have to worry about series that don’t converge, and actually do the series convergence tests and all that fun stuff. In physics, particularly low-energy experimental physics, you can usually just dive into working out the terms. If the series doesn’t converge, it generally becomes obvious pretty quickly when your results bear no resemblance to reality, and then you know you have to do something else. Like go to Wall Street, and construct complicated financial instruments that bear no resemblance to reality…
Pure math doesn’t have reality as an ultimate test, so you have to be a lot more careful.