If the series $\sum_{n=1}^\infin |a_n|$ converges, then $\sum_{n=1}^\infin a_n$ also converges.

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one way, so $\sum_{n=1}^\infin a_n$ can converges while $\sum_{n=1}^\infin |a_n|$ diverges, just like nth term

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Such a series is called absolutely convergent. Notice that if it converges on its "own," the alternator only allows it to converge more "rapidly."

$\sum_{n=1}^\infin a_n$ is conditionally convergent if $\sum_{n=1}^\infin a_n$ converges but $\sum_{n=1}^\infin |a_n|$ diverges.