Constructing a divergent series Please would you help me with this question? I've been thinking about it for ages but I've made very little headway, so if possible a hint would be ideal. 
Let  $\sum_{n=1}^∞{x_n}$ be a divergent series, where $x_n > 0$ for all $n$. Show that there is a divergent series  $\sum_{n=1}^∞{y_n}$ with $y_n > 0$ for all $n$, such that $(\frac{y_n}{x_n}) → 0.$
I have not been taught analysis formally, hence my lack of progress. I know to consider the series as a sequence of partial sums, and I tried to take the contrapositive of the statement but that just overcomplicated matters. I know I don't have many ideas to present but I have been trying this for days.
Thank you in advance. 

Based on this result:
If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well
Let's take $S_n=\sum\limits_{k=1}^n x_n\to+\infty\quad$ then $\quad\displaystyle y_n=\frac{x_n}{S_n}$ agrees with your requirements.


*

*$x_n>0\implies S_n>0\implies y_n>0$

*$\displaystyle \frac{y_n}{x_n}=\frac 1{S_n}\to 0$

*$\sum\ y_n$ diverges

For instance the classical divergent series $x_n=\frac 1n\ $ gives $\  y_n\sim\frac 1{n\ln(n)}$ 
