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**dori1123** Let $\displaystyle a_n, b_n$ be nonzero complex numbers. Suppose $\displaystyle lim_{n\to\infty}|\frac{a_n}{b_n}|=l$ exists, and $\displaystyle l \not= 0, \infty$.

1. Show that if one of the series $\displaystyle \sum_{n=0}^\infty a_n$ and $\displaystyle \sum_{n=0}^\infty b_n$ converges absolutely, then so does the other.

2. What if $\displaystyle l=0$ or $\displaystyle l=\infty$?

For #1, I assume $\displaystyle \sum_{n=0}^\infty a_n$ converges absolutely and trying to prove $\displaystyle \sum_{n=0}^\infty b_n$ converges absolutely. That is, $\displaystyle \forall \epsilon>0$ there exists $\displaystyle N \in \mathbb{N}$ such that $\displaystyle |b_{n+1}|+|b_{n+2}|+...+|b_{n+p}|<\epsilon$ for $\displaystyle p>0$.

I tried to use what I have to prove this, but not getting what I want. Can I get some help?

I don't know #2...