We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l }
b _ { 0 } = - 1 \\
\forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k }
\end{array} \right.$$
Prove that, for all $n$ in $\mathbb { N } ^ { * } , \left| b _ { n } \right| \leqslant 1$. Deduce an inequality on the radius of convergence of the power series $\sum _ { k \geqslant 0 } b _ { k } t ^ { k }$.