Proof for Absolute Convergence

Very first notice that |a_n| is either a_n or it is - a_n depending upon its sign.  The meaning of this is that we can then say,

0 < a_n +| a_n| < 2 |a_n|

Now here, as we are assuming that ∑|a_n| is convergent then ∑ 2|a_n| is as well convergent since we can just factor the 2 out of the series and 2 times a finite value will still be finite.  Though this permits us to use the Comparison Test to say that ∑a_n + |a_n| is as well a convergent series.

 At last, we can write,

 ∑a_n = ∑ a_n + | a_n| - ∑ |a_n|                                                     

and thus ∑a_n is the difference of two convergent series and thus is also convergent.

Fact about Absolute Convergence

If ∑a_n is absolutely convergent then it is as well convergent.