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Definition
If limn→∞an does not exist or if limn→∞an ≠ 0, then the series ∑an is divergent. |
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Definition
(a) If ∫ƒ(x)dx is convergent, then ∑an is convergent.
(b) If ∫ƒ(x)dx is divergent, then ∑an is divergent. |
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Definition
| The p-series ∑1/np is convergent if p>1 and divergent if p≤1. |
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| If limn→∞an/bn=c where 0<c<∞, then either both series converge or both series diverge. |
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(i) bn+1≤bn for all n
(ii) limn→∞bn=0
then the series is convergent. |
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Test for Absolute Convergence |
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Definition
| If the ∑|an| converges, then the ∑an absolutely converges. |
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Definition
limn→∞|an+1/an|=L:
(i) If L<1, then the series ∑an is absolutely convergent.
(ii) If L>1 or L=∞, then the series ∑an is divergent.
(iii) If L=1, inconclusive. |
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