# Divergence of absolute series

1. Nov 13, 2008

### musashi1029

1. The problem statement, all variables and given/known data

Show that following statement is true:
If Σa_n diverges, then Σ|a_n| diverges as well.

2. Relevant equations

Comparison Test:
If 0 ≤ a_n ≤ b_n for all n ≥ 1, and if Σa_n diverges, then Σb_n diverges as well.

3. The attempt at a solution

I tried to prove the statement by using the Comparison Test with a_n = a_n and b_n = |a_n|, but the condition for the Comparison Test is that both sequences must be greater than or equal to zero, which is not true for this problem. I would like to know if using the Comparison Test is a right approach to prove this statement.

Think of the Cauchy criterion; that is, $\sum a_n$ converges if and only if for all $\epsilon >0$ there exists an integer N such that $\left| \sum_{k=n}^ma_k \right| \leq \epsilon$ if $m\geq n\geq N$.
$$\left| \sum_{k=n}^ma_k \right| \leq \sum_{k=n}^m |a_k|$$