Weierstrass Theorem for Closed Domains I have an infinite sequence of complex functions $( f_n(z) )_{n \geq 1}$ each holomorphic on an open connected set $U \subset \mathbb{C}$. We know by Weierstrass' Theorem that if $f_n$ converges to a function $f$ (also defined on $U$) on every compact subset $K$ of $U$, then $f$ is also holomorphic on $U$. I was wondering what happens if we replace the condition "open" by "closed". More precisely, can we conclude the same when $U$ is the closed region $\{z \in \mathbb C:\Re(z) \geq 1\}$?
 A: The answer is no, you only can say that $f$ is continuous on $\Re z =1$ but nothing about its holomorphicity there  - for example, take $f(s)=\sum_{n \ge 2}\frac{1}{n^s\log^2n}$ and $f_N$ the partial series up to $n=N$ so $f_N$ are entire functions; $f_N$ converges absolutely (hence normally) to $f$ including on the line $\Re s=1$ but $f$ has a singularity at $1$ (either by Landau theorem since $1$ is its abscissa of convergence or just noting that if it were holomorphic in a neighborhood of $1$, its second derivative would be...) 
Taking enough powers of the logarithms in the denominator one can make $f$ arbitrarily smooth (in the real sense as a function of $t, s=1+it$, on the boundary line and with more care one should be able to even make it infinitely differentiable (in the real sense) on the boundary line 
Note also that if you take a function that has a wall of poles on $\Re s=1$ so it is not extendable at any point on the line and repeat the procedure (ie take a high enough integral to ensure convergence of the partial sums on the boundary) you get examples which are not holomorphic at any point on the boundary line
A: The answer by Conrad already explains how this fails for $U$. Let me add a few famous examples where we consider the closed unit disc instead.


*

*The partial sums of $\sum \frac{z^{2^n}}{2^n}$ converge absolutely on $\overline{\mathbb{D}}$ and the function is continuous there, but it is not holomorphic at any point on $\partial \mathbb{D}$ (actually, but this is longer to prove, the function is not even differentiable on $\partial \mathbb{D}$)

*The partial sums of $\sum \frac{z^{2^n}}{2^{nk}}$ converge absolutely on $\overline{\mathbb{D}}$ and the function is $C^{k-1}$ on $\partial \mathbb{D}$, but it is not holomorphic at any point on $\partial \mathbb{D}$

*The partial sums of $\sum \frac{z^{2^n}}{n!}$ converge absolutely on $\overline{\mathbb{D}}$ and the function is $C^{\infty}$ on $\partial \mathbb{D}$, but it is not holomorphic at any point on $\partial \mathbb{D}$
Proving that the examples are not holomorphic on $\partial\mathbb{D}$ is quite easy for the first two (hint: consider the $k$-th derivative). For the third one, you need the Hadamard-Ostrowski gap theorem, which is a general tool for this kind of series (namely, lacunary series).
