Does Divergence of a Series Imply Divergence of Its Absolute Values?

[musashi1029]

Homework Statement

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

Homework Equations

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

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.

Thank you in advance.

 

[jeffreydk]

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.

Now think of what you have to prove and the fact that

\left| \sum_{k=n}^ma_k \right| \leq \sum_{k=n}^m |a_k|