Let us revisit the geometric progression sum considered in an earlier article,
where here is a complex number. For what values of does this infinite sum make sense? Can we find a closed-form expression for in such cases? To investigate this, we fix to some value and consider the partial sums:
where we just add the first terms of . Now if tends to a finite limit as (can we for any find an such that for all ?) then we have .
Let us first single out the special case . Since we cannot assign any well-defined, finite value to , so is divergent for . For we get
Let us consider three different cases. If we see that the only term that depends on tends to zero so we suspect that the limit is ,
Since the magnitude of the difference between our suspected limit and the partial sums can be made as small as we like (as long as we choose sufficiently large), we have
What about ? We get
and we see that as . We can thus not find a finite limit to which tends as , so the series is divergent for .
Left to consider is the case , , and this is where it gets interesting. We get
So the partial sums are bounded by some constant independent of . Does the value work as a limit in this case also? We set with and subtract,
(using and ). So does not converge to as . Indeed, we see that follows a circle in the complex plane; a circle centered in with radius . And this is what I find interesting: does not converge to any value,
but circles around the value when , . In fact, makes sense for all , so can this value be assigned to in some meaningful way? (When , I would suspect that the values of spirals inward towards as grows and spirals outwards when ; I have not verified this, though.)
This reminded me that G. H. Hardy has written a book called Divergent Series, where he manipulates infinite series with an “entirely uncritical spirit”. Therein, he also considers the series and, e.g., can somehow make sense. I have only flicked through the book (excerpt), but I think I should take a closer look…