I was thinking about slowly diverging series. I came up with this idea but I don't know what it's called or where to look.
Suppose you start with the harmonic series:
$$\sum_1^{\infty}\frac{1}{n} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots$$
this diverges fairly slowly. Now we remove geometric series, seriatim. We know that a geometric series $\sum_{n=1}^{\infty}ar^n$ converges if $|r| < 1$. So, remove first $1 + \frac{1}{2} + \frac{1}{4} + \cdots$ leaving:
$$\frac{1}{3} + \frac{1}{5} + \frac{1}{6} + \cdots$$
then remove $\frac{1}{3} + \frac{1}{9} + \cdots $, leaving
$$\frac{1}{5} + \frac{1}{6} + \cdots$$
and so on. That is, after each removal, we remove a geometric series that begins with the first term that remains in the series. So, we next remove $\frac{1}{5} + \frac{1}{25} + \cdots $, then $\frac{1}{6} + \frac{1}{36} + \cdots$
Since we are removing convergent series, no matter how many we remove (as long as its not uncountably many), the result is still divergent.
Has this series been studied?