Uniform convergence on the closed unit disc does not give a holomorphic extension to a larger disc

Given: The complex power series defining $F$, with its partial sums and convergence interpreted as in Complex series, absolute convergence, complex power series, and radius of convergence.

If $\mathrm{\mid }{f}_n(x)\mathrm{\mid }\le M_n$ and the real series $\sum M_n$ converges, then the complex function series $\sum f_n$ converges absolutely pointwise and uniformly (Weierstrass M-test for complex-valued function series).

The series ${\sum }_{n\ge 1}1\mathrm{/}{n}^2$ converges, while the harmonic series ${\sum }_{n\ge 1}1\mathrm{/}n$ diverges (For rational $p>0$, $\sum 1\mathrm{/}{k}^p$ converges iff $p>1$).

Inside the radius of convergence, a complex power series may be differentiated term by term: $({\sum }_{n\ge 0}{c}_n{z}^n{)}^{\mathrm{\prime }}={\sum }_{n\ge 1}n{c}_n{z}^{n-1}$ (Inside its disc of convergence a complex power series is holomorphic and may be differentiated term by term).

A continuous real-valued function on a nonempty compact metric space is bounded and attains a maximum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

A complex differentiable function is continuous (Complex differentiability at a point implies continuity there).

Complex polynomials, including every monomial, are holomorphic (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero).

Closed bounded subsets of Euclidean space, including the interval $[0,1]$, are compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Complex modulus is multiplicative and satisfies the triangle inequality, hence $\mathrm{\mid }\mathrm{\mid }u\mathrm{\mid }-\mathrm{\mid }v\mathrm{\mid }\mathrm{\mid }\le \mathrm{\mid }u-v\mathrm{\mid }$ (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

Every holomorphic function has complex derivatives of every order locally (All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle).