If $S_n\to \infty$ as $n\to \infty$ is the following inequality valid or when is it valid? $\frac{a_n}{S_{n-1}}\leq \frac{C}{n}.$

Question

Let $a_n$ be a sequences of positive real numbers and $S_n=\sum_{k=1}^{n}a_k.$

If $S_n\to \infty$ as $n\to \infty$ is the following inequality valid or when is it valid?$(C>0)$ $$\frac{a_n}{S_{n-1}}\leq \frac{C}{n}.$$

Answer 1

Hint: Consider the case $a_n=2^n$