cal 2 spr12 oversteegen Series Testing - Series Testing Section 8.2 The Test for Divergence If limn-> An does not exist or if limn-> An 0 then the

cal 2 spr12 oversteegen Series Testing - Series Testing...

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Series Testing Section 8.2The Test for Divergence:If limn-> ∞ An does not exist or if limn->∞ An ≠ 0, then the series ∑ Adivergent.TheoremhjIf ∑ Anand ∑ Bnare convergent series, then so are the series ∑ CAn(where C is a constant), ∑ (An+Bn), and ∑ (An-Bn), and (i)∑ CAn= C ∑ An(ii)∑ (An+ Bn)= ∑ An+ ∑ Bn(iii)∑ (An- Bn)=∑ An - ∑ BSection 8.3The Intergral Test Suppose fis a continuous, positive, decreasing function on [1,∞) and let An= f(n). Then the series ∑ Anis convergent if and only if the improper intergral ∫f(x)dx is convergent. In other words:(a)If f(x) dx is convergent, then ∑ An is convergent.(b)If f(x) dx is divergent, then ∑ Anis divergent.P- seriesThe p-series ∑ 1/npis convergent if p> 1 and divergent if p≤ 1.The Comparison TestSuppose that ∑ Anand ∑ Bnare series with positive terms.(a)If ∑ Bnis convergent and An≤ Bnfor all n, then ∑ Anis also convergent.(b)If ∑ Bnis divergent and An≥ Bnfor all n, then ∑ Anis also divergent.The Limit Comparison TestSuppose that ∑ A n is n

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