Absolutely Convergent Series

Topics: Calculus - Series - Sequence

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(definition)

We say that a series ${\sum }_{n=1}^{\infty }{a}_n$ is absolutely convergent if and only if ${\sum }_{n=1}^{\infty }|a_n|$ converges.

(theorem)

If $\sum a_n$ is absolutely convergent, then it’s also convergent.

If the series of the absolute values of the negative terms and the series of the positive terms are convergent, then $\sum a_n$ is absolutely convergent.

Leibniz Theorem §

(theorem)

Suppose that $a_1\ge a_2\ge a_3\ge \cdots \ge 0$, and that ${\lim }_{n\to \infty }{a}_n=0$.

Then, ${\sum }_{n=1}^{\infty }(-1{)}^{n+1}{a}_n=a_1-a_2+a_3-a_4+\dots$ converges.