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) $\sum_{k=3}^{\infty} \frac{1}{\sqrt{b(\log k)}} < \infty$