Power series summation equation

[vibhuav]

(Was posted in general physics forum also)

I am currently reading Roger Penrose’s “Road to Reality”. In section 4.3, Convergence of power series, he refers to the sum of the series:
1 + x² + x⁴ + x⁶ + x⁸ + ... = 1/(1-x²)

Of course, this is true for |x| < 1, beyond which the series diverges and the equation for the summation does not apply. Penrose uses this equation and the similar one, namely 1 - x² + x⁴ - x⁶ + x⁸ - ... = 1/(1+x²), to demonstrate the “reality” of imaginary numbers. I understood that and the argument is very impressive.

As a digression, he uses the first equation for x=2:
1 + 2² + 2⁴ + 2⁶ + 2⁸ + ... = 1/(1-2²) = -1/3

which is obviously nonsensical. But then, he goes on to say “Yet, I think that it is important to appreciate that, in the appropriate sense, Euler really know what he was doing when he wrote down apparent absurdities of this nature, and that there are senses according to which the above equation must be regarded as ‘correct’...”. Further on, he says “...In fact, it is perfectly possible to give a mathematical sense to the answer ‘-1/3’ to the above infinite series, but one must be careful about the rules telling us what is allowed and what is not allowed. [refers the reader to GH Hardy’s book, Divergent Series]” and then he says “in modern physics, particularly in the areas of QFT, divergent series of this nature are frequently encountered...”

In what sense is the above absurdity ‘correct’? Can someone elaborate for me? I understand the part that outside the circle of convergence, one cannot reach the answer (infinity) by attempting to sum the series directly; such an attempt keeps diverging from the answer provided by 1/(1-x²). But it the ‘correctness of -1/3’ that I am not able to understand.

 

[The_Duck]

A key idea here is known as analytic continuation. There is a region of values for x in which the series S(x) = 1 + x^2 + x^4 + ... can be summed straightforwardly to get a finite answer. In this region S(x) defines a perfectly well behaved function of x. There is a nice result in complex analysis that any sufficiently nice function defined on some region of the complex plane can be extended ("continued") to a function defined on the entire complex plane, and this extended function is essentially unique. 1/(1 - x^2) is the extended version of the function S(x): it agrees with S(x) for |x| < 1, and it is defined on the entire complex plane (excluding 1 and -1, where it blows up).

Maybe a better way to think of it is that, if we find ourselves using the equation "1 + 2^2 + 2^4 + ... = -1/3," we were never really interested in literally summing the series 1 + 2^2 + 2^4 + ... . Rather, we were interested a function f(x) which happens to agree with the series S(x) = 1 + x^2 + x^4 + ... when that series converges, but which is also defined outside that region of convergence. That function, f(x) = 1/(1 - x^2), is the true object of our study. From this perspective, the equation 1 + 2^2 + 2^4 + ... = -1/3 is rather silly. Yes, for |x| < 1 we can represent f(x) by the power series 1 + x^2 + x^4 + ..., but for |x| > 1 we can't really, and it's perverse to write an equation suggesting that you can.

 

[JJacquelin]

Hi vibhuav !

one must not forget a term in the closed form.
If the forgotten term tends to 0, then OK, the closed form is correct.
If the forgotten term tends to infinity, then obviously the closed form is unsuitable.