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(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 + x

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 - x

As a digression, he uses the first equation for x=2:

1 + 2

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-x

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 + x

^{2}+ x^{4}+ x^{6}+ x^{8}+ ... = 1/(1-x^{2})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 - x

^{2}+ x^{4}- x^{6}+ x^{8}- ... = 1/(1+x^{2}), 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 + 2

^{2}+ 2^{4}+ 2^{6}+ 2^{8}+ ... = 1/(1-2^{2}) = -1/3which 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-x

^{2}). But it the ‘correctness of -1/3’ that I am not able to understand.
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