Exam 1 - 1 THE TEST FOR DIVERGENCE If lim an does not exist or if n lim an = 0 then the series n an is divergent n=1 THE INTEGRAL TEST Suppose is a

Exam 1 - 1 THE TEST FOR DIVERGENCE If lim an does not exist...

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1 THE TEST FOR DIVERGENCE If lim n →∞ a n does not exist or if lim n →∞ a n 6 = 0, then the seriesXn=1anis divergent.THE INTEGRAL TESTSuppose is a continuous, positive, decreasingfunction on [1,) and letan=f(n) . Then the seriesXn=1anis convergent ifand only if the improper integralZ1f(x)dxis convergent. In other words:(i) IfZ1f(x)dxis convergent, thenXn=1anis convergent.(ii) IfZ1f(x)dxis divergent, thenXn=1anis divergent.THE COMPARISON TESTSuppose thatXn=1anandXn=1bnare serieswith positive terms.(i) IfXn=1bnis convergent andanbnfor alln, thenXn=1anis also convergent. (ii) Ifbnis divergent andanbnfor alln, thenanis also divergent.
2 THE ALTERNATING SERIES TEST If the alternating series X n =1 ( - 1) n - 1 b n = b 1 - b 2 +