Best guess for the function is

By the ratio test, the series converges for

When
,
is a convergent
-series.
When
,
is a convergent alternating series.
So, the interval of convergence for
is the <em>closed</em> interval
.
The derivative of
is the series

which also converges for
by the ratio test:

When
,
becomes the divergent harmonic series.
When
,
is a convergent alternating series.
The interval of convergence for
is then the <em>closed-open</em> interval
.
Differentiating
once more gives the series

The first series is geometric and converges for
, endpoints not included.
The second series is
, which we know converges for
.
Putting these intervals together, we see that
converges only on the <em>open</em> interval
.