Limit of the ratio of two divergent sequences

Question

Let $∑a_n $ be a series of real numbers which converges, but not absolutely.

Let $ p_n = \frac{a_n +|a_n|}{2} \ge 0, \ q_n = \frac{a_n - |a_n|}{2} \ \le 0 $.

Let $ P_n = \sum_{k=1}^{n}{p_k}, \ Q_n = \sum_{k=1}^{n}{q_k} $.

Show that ${ \lim_{n\to\infty} \frac{Q_n}{P_n} } = -1$.

I was able to show that both $ {Q_n} $ and $ {P_n} $ diverges. But I couldn't proof the ratio between them (I'm not sure if it's helpful). How can it be proofed?

Answer 1

Note that $$P_n + Q_n = \sum_{k=1}^n a_k$$ so $$ 1+ \frac{Q_n}{P_n} = \frac{\sum_{k=1}^n a_k}{P_n} \rightarrow \frac{const.}{\infty} = 0 $$