Let (a_n) be an enumeration of all complex numbers having rational
real and imaginary parts, and modulus (strictly) greater than one,
i.e., all elements of Q[i] not contained in the closed unit disk.

Let (c_n) be a decreasing sequence of real numbers which converges
rapidly toward zero.  (I'm deliberately being vague on what exactly I
mean by "rapidly", but I'll say more in a few lines.)

Let F(z) be the sum of c_n/(z-a_n) for all n, provided this series
converges.  We let f(z)=F(z) for z in the open unit disk.

If c_n is chosen to decrease fast enough, then the series converges
uniformly on the closed unit disk.  Then F is continuous on the closed
unit disk and its restriction f to the open unit disk is holomorphic.
At the very least, we shall assume (c_n) such that the series
converges uniformly on every compact set within the open unit disk
(so, again, f is holomorphic on the open unit disk).

The question is then: show (or refute) that f does not admin a
holomorphic extension to any larger (connected) open set (that is, the
unit circle is the natural boundary of holomorphy of f).

Intuitively the reason seems clear: we have created a "pole" at every
rational point outside the closed unit disk, so we cannot get a
holomorphic extension there (nor even a meromorphic extension, since
the poles would be dense).  I have been unable, however, to make this
reasoning rigorous.

Note that if (c_n) decreases rapidly enough then f and all its
derivatives extend continuously to the closed unit disk.  One might
hope to prove that the derivatives at a point on the unit circle
increase too rapidly, but I do not see why this should be.

It is infuriating that the series for F converges - at least when
(c_n) decreases rapidly enough - for many points outside the unit
disk: in fact, on the complement H of a (dense) set of Lebesgue
measure zero.  It is probably easy to see that F on H, defined as the
sum of the converging series, cannot be extended to a holomorphic
function on any open set not contained in the open unit disk: however,
I see no reason why a holomorphic extension of f should indeed
coincide with the sum F of the series outside the unit disk, so this
line of attack seems doomed.

Did I miss a simple fact from complex analysis?  

I would welcome any thoughts on this problem.  Thanks in advance!