# janmr blog

## On the Divergence of a Geometric Progression Sum28 August 2009

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