If a series grows more slowly than any geometric series, can it ever converge to a rational?

Question

I was reading a proof of $e$'s irrationality which, in some sense, uses the fact that the series $\sum \frac{1}{n!} = e$ grows slowly.

This got me thinking: can we generalize this and say "oh, $\sum \frac{1}{n!}$ has a inverse factorial rate of growth, that's too slow, hence it can't converge to a rational".

The closest in the rate-of-growth hierarchy I can think of are the geometric series, which do converge to rationals. So that was my guess of where do draw the line. Does there exist a function which a) grows more slowly than any geometric function, and b) converges to a rational number?

If there is one, is there such a function that grows as slow or slower than $\sum \frac{1}{n!}$?

Answer 1

One could say that $\sum0=0$, it grows slower than both of your conditions and it is a rational number.

Also, $\sum\frac{[\ln(2)]^n}{n!}=2$, it is both a rational number, and it grows slower than a geometric sequence.

I'm not quite sure if I know any that grow slower than your factorial problem, but if it did, it could easily be tweaked to produce a rational number.

Answer 2

The question makes no sense, really. Take any convergent series; take its limit $l$, and if the limit is not rational, add something just to the first term so the new limit is rational. (For example, add $-l$ to the first term to make the sum of the series equal to $0$.)

Did you mean something else, such as "all the terms are rational"? And perhaps also "strictly positive"?