The series diverges. A common way to make it convergent is to replace each with a power of ; the new series will converge when and maybe its sum will have a limit as . And indeed,
which tends to as approaches from the left.
Things get more interesting if instead of consecutive integers as exponents, we use consecutive powers of :
On most of the interval it behaves just like the previous one:
But there appears to be a little blip near . Let’s zoom in:
And zoom in more:
This function was considered by Hardy in 1907 paper Theorems concerning infinite series. On pages 92–93 he shows that it “oscillates between finite limits of indetermination for “. There is also a footnote: “The simple proof given above was shown to me by Mr. J. H. Maclagan-Wedderburn. I had originally obtained the result by means of a contour integral.”
Okay, but what are these “finite limits of indetermination”? The Alternating Series Estimation shows for , but the above plots suggest that oscillates between much tighter bounds. Let’s call them and .
Since , it follows that as . Hence, . In other words, and are symmetric about . But what are they?
I don’t have an answer, but here is a simple estimate. Let and observe that
The function is not hard to understand: its graph is a parabola.
Since is positive on , any of the terms in the sum (1) gives a lower bound for . Each individual term is useless for this purpose, since it vanishes at . But we can pick in depending on .
Let be the unique solution of the equation . It could be written down explicitly, but this is not a pleasant experience; numerically . For every there is an integer such that , namely the smallest integer such that . Hence,
which gives a nontrivial lower bound and symmetrically . Frustratingly, this falls just short of neat and .
One can do better than (2) by using more terms of the series (1). For example, study the polynomial and find a suitable interval on which its minimum is large (such an interval will no longer be symmetric). Or use consecutive terms of the series… which quickly gets boring. This approach gives arbitrarily close approximations to and , but does not tell us what these values really are.