Let $\sum _ { n \geqslant 0 } a _ { n } z ^ { n }$ be a power series with radius of convergence $R \geqslant 1$ and sum $f$. We denote $$\Delta _ { \theta _ { 0 } } = \left\{ z \in \mathbb { C } ; | z | < 1 \text { and } \exists \rho > 0 , \exists \theta \in \left[ - \theta _ { 0 } , \theta _ { 0 } \right] , z = 1 - \rho e ^ { i \theta } \right\}$$ for $\theta _ { 0 } \in [ 0 , \pi / 2 [$.
The purpose of this question is to prove that $$\left( \sum _ { n \geqslant 0 } a _ { n } \text { converges } \right) \Rightarrow \left( \lim _ { \substack { z \rightarrow 1 \\ z \in \Delta _ { \theta _ { 0 } } } } f ( z ) = \sum _ { n = 0 } ^ { + \infty } a _ { n } \right) \qquad \text{(Abel)}$$
(a) Prove (Abel) for $R > 1$.
From now on, we assume that $R = 1$ and that $\sum _ { n \geqslant 0 } a _ { n }$ converges, and we are given a $\theta _ { 0 } \in [ 0 , \pi / 2 [$.
(b) Prove that for all $N \in \mathbb { N } ^ { * }$ and $z \in \mathbb { C } , | z | < 1$, we have $$\sum _ { n = 0 } ^ { N } a _ { n } z ^ { n } - S _ { N } = ( z - 1 ) \sum _ { n = 0 } ^ { N - 1 } R _ { n } z ^ { n } - R _ { N } \left( z ^ { N } - 1 \right)$$
(c) Deduce that for all $z \in \mathbb { C } , | z | < 1$, we have $$f ( z ) - S = ( z - 1 ) \sum _ { n = 0 } ^ { + \infty } R _ { n } z ^ { n }$$
(d) Let $\varepsilon > 0$. Prove that there exists $N _ { 0 } \in \mathbb { N }$ such that for all $z \in \mathbb { C } , | z | < 1$ $$| f ( z ) - S | \leqslant | z - 1 | \sum _ { n = 0 } ^ { N _ { 0 } } \left| R _ { n } \right| + \varepsilon \frac { | z - 1 | } { 1 - | z | }$$
(e) Prove that there exists $\rho \left( \theta _ { 0 } \right) > 0$ such that for all $z \in \Delta _ { \theta _ { 0 } }$ of the form $z = 1 - \rho e ^ { i \theta }$ with $0 < \rho \leqslant \rho \left( \theta _ { 0 } \right)$, we have $$\frac { | z - 1 | } { 1 - | z | } \leqslant \frac { 2 } { \cos \left( \theta _ { 0 } \right) }$$ Deduce (Abel).