Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence. We assume that the associated power series $\sum a _ { n } x ^ { n }$ has radius of convergence $R _ { a } = 1$ and that the sum $f$ of this series satisfies $f ( x ) \sim \frac { 1 } { 1 - x }$ when $x \rightarrow 1, x < 1$. Determine a real sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ such that $$\forall x \in ] - 1,1 [ , \quad \frac { 1 } { 1 - x ^ { 2 } } = \sum _ { n = 0 } ^ { + \infty } b _ { n } x ^ { n }$$
Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence. We assume that the associated power series $\sum a _ { n } x ^ { n }$ has radius of convergence $R _ { a } = 1$ and that the sum $f$ of this series satisfies $f ( x ) \sim \frac { 1 } { 1 - x }$ when $x \rightarrow 1, x < 1$.
Determine a real sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ such that
$$\forall x \in ] - 1,1 [ , \quad \frac { 1 } { 1 - x ^ { 2 } } = \sum _ { n = 0 } ^ { + \infty } b _ { n } x ^ { n }$$