8
$\begingroup$

We say $a_n$ converges slower than $b_n$ if there exist an $x$ such that for all $m>x$, $a_m>b_m$ and both $\sum a_n$ and $\sum b_n$ converges.

Ignoring constant factors, which type of function converges the slowest?

  • 0
    See also Qiaochu Yuan's answer to this related question: http://math.stackexchange.com/questions/9350/smallest-function-whose-inverse-converges/9353#93532011-02-04

1 Answers 1

16

There is not such a function! Walter Rudin in his "Principles of Mathematical Analysis" has a series of exercises trying to indicate that there is no function at the "boundary" between convergence and divergence.

There was an interesting series of answers about this very issue at MathOverflow (MO): https://mathoverflow.net/questions/49415/nonexistence-of-boundary-between-convergent-and-divergent-series

A quick way of seeing that there is no "slowest" convergent series is Rudin's exercise 12.b, mentioned at the link above: If $\sum a_n$ converges, and the $a_n$ are positive, then $\sum a_n/\sqrt {r_n}$ converges, where $r_n$ is the tail $\sum_{i\ge n}a_i$. Note $r_n\to 0$, so $a_n/\sqrt{r_n}>a_n$ for $n$ large enough.

I recommend that you take a look at the answers at MO and at the references they suggest, for more subtle examples.

  • 0
    @StevenStadnicki I know this is from nearly six years ago, but is there any chance you ever found where this is from or are now willing to look for it? I'm very interested in this 'primitive recursive boundary', but the only google result for it is this comment.2016-12-23