Given two complex sequences $(a_n)_{n_\in\mathbb{N}},(b_n)_{n\in\mathbb{N}}\subseteq \mathbb{C}$ that converge to zero we consider the series $\sum a_n$ and $\sum b_n$. Assume now that the first converges and the second diverges to infinity. It feels fairly obvious (but apparently isn't at all) that $$\frac{a_n}{b_n}\to 0 \text{ as }n\to\infty.$$ Nonetheless I have failed to prove this, probably because I am quite rusty on the basics and missing something hugely obvious. Can anyone help me out? Do I need absolute convergence of the first series?
2026-04-03 17:53:06.1775238786
Convergence speed of generating sequences for potentially convergent series
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