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

Discussion [Using Flash] [Using Java]

Exercises:

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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.

• Problem. Find an approximation of the series
  

  using the partial sum s₁₀₀. What is the maximum possible error using this approximation?
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• Problem. Find an integer n such that, using s_n as an approximation of the series
  

  the maximum possible error is at most .0001.
  [Solution.]

• Drill Problems on using the Alternating Series Test.