Other Convergence Tests CHAPTER 2 2.4 Continuity The Alternating Series Tests: If the alternating series  n=1 (-1)n-1bn = b1 – b2 + b3 – b4 + … bn >0 satisfies (a) bn+1  bn for all n (b) lim n   bn = 0 Then the series is convergent.

CHAPTER 2 The Alternating Series Estimation Theorem: If s =  (-1) n-1 bn is the sum of an alternating series that satisfies (a) 0 < bn+1  bn and (b) lim n   bn = 0 then | Rn | = | s – sn |  bn+1. 2.4 Continuity Definition: A series  an is called absolutely convergent if the series of absolute values  | an | is convergent. Theorem: If a series  an is absolutely convergent, then it is convergent.

CHAPTER 2 2.4 Continuity The Ratio Test: If lim n   | (an+1) / an | = L < 1,then the series  n=0 an is absolutely convergent (and therefore convergent). If lim n   | (an+1) / an | = L > 1 or lim n   | (an+1) / an | = , then the series  n=0 an is divergent . 2.4 Continuity