# Limit of the ratio of two divergent sequences

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?

Note that $$P_n + Q_n = \sum_{k=1}^n a_k$$ so $$1+ \frac{Q_n}{P_n} = \frac{\sum_{k=1}^n a_k}{P_n} \rightarrow \frac{const.}{\infty} = 0$$
• We just subtract 1 from both sides. The limit of a sum is a sum of the limits, in this case $$0 = \lim_{n\rightarrow\infty}(1+\frac{Q_n}{P_n}) = (\lim_{n\rightarrow\infty}1)+(\lim_{n\rightarrow\infty}\frac{Q_n}{P_n}) = 1+ \lim_{n\rightarrow\infty}\frac{Q_n}{P_n}$$ May 8 '19 at 10:32
• Thanks! It might be out of this question's scope, but how do we show that $P_n$ necessarily goes to infinity? Is it enough to say the the series $P_n$ is monotonically increasing and diverges? May 8 '19 at 13:43