If $a_n \in \Bbb R$ s.t. $|a_n|$ diverges while converging conditionally.Then prove that the series ${a_n}^+ $ diverges.

Question

I was trying to solve the following problem :

Let $a_n \in \Bbb R$ s.t. $\sum_{n=1}^\infty {|a_n|} = \infty$ and $\sum_{n=1}^m {a_n} \to a \in \Bbb R$ as $m \to \infty$. Now let, ${a_n}^+ = max\{a_n , 0\}$ . Then prove that $\sum_{n=1}^\infty {a_n}^+ = \infty$ .

Clearly, $ a_n \leq {a_n}^+ \leq |a_n| , \forall n \in \Bbb N$ . But how to move forward? Thanks in advance for help.

Answer 1

Hint: $${}{}{} x^+=\frac{|x|+x}2 $$

Answer 2

Hint:

We are given that $\sum_{k=1}^n a_k = \sum_{k=1}^n a_k^+ - \sum_{k=1}^n a_k^- $ converges.

If $\sum_{k=1}^n a_k^+$ converges then so does $\sum_{k=1}^n a_k^- = \sum_{k=1}^n a_k^+ - \sum_{k=1}^n a_k $.

Note that $|a_k| = a_k^+ + a_k^-$.