Show that for all $k \in \mathbf { N }$, the real numbers $b _ { k } = \sum _ { n = 1 } ^ { + \infty } \lambda _ { n } ^ { k } a _ { n }$ are well-defined.
The sequences satisfy: $\left| a _ { n } \right| \leq \frac { M } { 2 ^ { n } }$ for some $M \in \mathbf{R}_+^*$, and $\lambda_n$ is strictly increasing with $\lambda_0 = 0$, $\lim_{n\to+\infty} \lambda_n = +\infty$, and $\lambda_n \underset{n\to+\infty}{=} O(n)$.

For every increasing function $b : [1, \infty) \rightarrow [1, \infty)$ such that $$\int_1^\infty \frac{\mathrm{d}x}{b(x)} < \infty$$ we must have
(A) $\sum_{k=1}^{\infty} \frac{\sqrt{\log k}}{b(k)} < \infty$
(B) $\sum_{k=3}^{\infty} \frac{\log k}{b(\log k)} < \infty$
(C) $\sum_{k=1}^{\infty} \frac{e^k}{b\left(e^k\right)} < \infty$
(D) $\sum_{k=3}^{\infty} \frac{1}{\sqrt{b(\log k)}} < \infty$