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Section 10.9 : Absolute Convergence

1. Determine if the following series is absolutely convergent, conditionally convergent or divergent.

\[\sum\limits_{n = 2}^\infty {\frac{{{{\left( { - 1} \right)}^{n + 1}}}}{{{n^3} + 1}}} \]

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Okay, let’s first see if the series converges or diverges if we put absolute value on the series terms.

\[\sum\limits_{n = 2}^\infty {\left| {\frac{{{{\left( { - 1} \right)}^{n + 1}}}}{{{n^3} + 1}}} \right|} = \sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \]

Now, notice that,

\[\frac{1}{{{n^3} + 1}} < \frac{1}{{{n^3}}}\]

and we know by the \(p\)-series test that

\[\sum\limits_{n = 2}^\infty {\frac{1}{{{n^3}}}} \]


Therefore, by the Comparison Test we know that the series from the problem statement,

\[\sum\limits_{n = 2}^\infty {\frac{1}{{{n^3} + 1}}} \]

will also converge.

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So, because the series with the absolute value converges we know that the series in the problem statement is absolutely convergent.