Let $\left( c _ { k } \right) _ { k \in \mathbb{N} }$ be a sequence of elements of $\mathbb{R}^{+}$ such that the power series $\sum c _ { k } x ^ { k }$ has radius of convergence 1 and the series $\sum c _ { k }$ diverges. Show that $$\sum _ { k = 0 } ^ { + \infty } c _ { k } x ^ { k } \underset { x \rightarrow 1 ^ { - } } { \longrightarrow } + \infty$$ With the element $A$ of $\mathbb{R}^{+*}$ fixed, one will show that there exists $\alpha \in ]0,1[$ such that $$\forall x \in ]1 - \alpha , 1[ , \quad \sum _ { k = 0 } ^ { + \infty } c _ { k } x ^ { k } > A$$
Let $\left( c _ { k } \right) _ { k \in \mathbb{N} }$ be a sequence of elements of $\mathbb{R}^{+}$ such that the power series $\sum c _ { k } x ^ { k }$ has radius of convergence 1 and the series $\sum c _ { k }$ diverges. Show that
$$\sum _ { k = 0 } ^ { + \infty } c _ { k } x ^ { k } \underset { x \rightarrow 1 ^ { - } } { \longrightarrow } + \infty$$
With the element $A$ of $\mathbb{R}^{+*}$ fixed, one will show that there exists $\alpha \in ]0,1[$ such that
$$\forall x \in ]1 - \alpha , 1[ , \quad \sum _ { k = 0 } ^ { + \infty } c _ { k } x ^ { k } > A$$