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Proof for Absolute Convergence
Very first notice that |an| is either an or it is - an depending upon its sign. The meaning of this is that we can then say,
0 < an +| an| < 2 |an|
Now here, as we are assuming that ∑|an| is convergent then ∑ 2|an| 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 ∑an + |an| is as well a convergent series.
At last, we can write,
∑an = ∑ an + | an| - ∑ |an|
and thus ∑an is the difference of two convergent series and thus is also convergent.
Fact about Absolute Convergence
If ∑an is absolutely convergent then it is as well convergent.
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