absolute_convergence

absolute_convergence - (12.6) Absolute Convergence We now...

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(12.6) Absolute Convergence We now are considering series whose terms might be positive or negative. But most of our convergence tests apply to series whose terms are positive only. To deal with this, we deal with the series of absolute values ± n =1 | a n | for a series ± n =1 a n . It turns out that if ± | a n | converges, so does ± a n .I f ± | a n | converges, we say the original series ± a n is absolutely convergent .I f ± a n converges but ± | a n | does not converge, we say ± a n is conditionally convergent . Example. ± n =1 ( - 1) n 1 n 2 . The series of absolute values is ± n =1 ² ² ² ² ( - 1) n 1 n 2 ² ² ² ² = ± n =1 1 n 2 . This is a p -series with p = 2, so it converges. Therefore the original series ± n =1 ( - 1) n 1 n 2 .converges absolutely. Example. ± n =1 ( - 1) n 1 n . The series of absolute values is ± n =1 ² ² ² ² ( - 1) n 1 n ² ² ² ² = ± n =1 1 n . This is the harmonic series (a p -series with p = 1), so it diverges. By the alternating series test, ± n =1 ( - 1) n 1 n converges. Therefore the original series ± n =1 ( - 1) n 1 n 2
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absolute_convergence - (12.6) Absolute Convergence We now...

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