# Convergence of infinite series for $\sum {{a_n}}$ and $\sum {{b_n}}$

Let $\sum {{a_n}}$ and $\sum {{b_n}}$ be two series of positive terms and suppose that the sequence {${a_n/b_n}$} converges to 0. Prove that

a) If $\sum {a_n}$ diverges to + $\infty$ then $\sum {b_n}$ diverges to +$\infty$

b) if $\sum {b_n}$ converges, then $\sum {a_n}$ converges

Hint: For any $C > 0$, we may choose an $N$ large enough so that the following holds: $$\sum_{n=N}^\infty b_n = \sum_{n=N}^\infty \frac{b_n}{a_n} a_n \geq \sum_{n=N}^\infty C a_n\\ \sum_{n=N}^\infty a_n = \sum_{n=N}^\infty \frac{a_n}{b_n} b_n \leq \sum_{n=N}^\infty C b_n$$ It suffices to consider $C=1$.

Take $\epsilon =1$, for enough great $n$,

$a_n =b_n \frac {a_n}{b_n}\leq b_n$

finish by comparison criterion.

Hint There exists $N$ such that if $n>N$, then $a_n<b_n$ by definition of the limit.