Theorem. The Alternating Series Test. Suppose that {ai} is a sequence of positive numbers such that
  1. ai > ai+1 for all i.

Then the series is convergent.

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Exercises:

Theorem. The Alternating Series Error Estimate. Suppose that {ai} is a sequence of positive numbers which satsifies the hypothesis of the theorem above. Suppose that the series

converges to L; let rn = L - sn where sn denotes the nth partial sum. Then |rn< an+1.

Discussion.