1. ## Convergent series

Show that if $a_n \geq 0$ and $\sum_{n=1}^{\infty}a_n<\infty}$, then we can find a $b_n$ sequence that $\frac{b_n}{a_n} \rightarrow \infty$, but $\sum_{n=1}^{\infty}b_n<\infty$ is true.
I.e for any convergent series, exists an asymptotically greater convergent series.

2. If q>0

$
\displaystyle
a_n \sim \frac{1}{n^{1+q}}
$

then

$
\displaystyle
b_n \sim \frac{1}{n^{1+q/2}}
$