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`sum_(n=1)^oo (-1)^(n+1)/(n+3)` Determine whether the series converges absolutely or conditionally, or diverges.

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pnrjulius eNotes educator | Certified Educator

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This is just the alternating harmonic series with the first three terms stripped out:

`sum_{n=1}^{infty} (-1)^{n+1}/{n+3} = sum_{n=4}^{infty} (-1)^{n-2}/{n}`
`= sum_{n=4}^{infty} (-1)^{n}/{n}`

We know that the alternating harmonic series converges conditionally, so this series also converges conditionally (because it is in fact the same series, with a constant subtracted).

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