Educator

Alternating Series

Main theorems:

Alternating Series Test: Suppose b_n > 0 for all n. Then ∑(−1)ⁿb_n (or ∑(−1)ⁿ⁺¹b_n ) is an alternating series. Check two conditions:

1. Is b_n decreasing? (i.e. is b_n+1 < b_n ?)

2. Is lim b_n = 0?

If both are true, then ∑(-1)ⁿb_n converges.

Estimates of sums: If ∑a_n is a series that satisfies AST, and we use the partial sum s_n as an estimate of the total sum S (the true answer), then our error E could be positive or negative, but it’s bounded by |a_n+1|:

|E| ≤ |a_n+1|

Hints and tips:

• You must have an alternating series for the Alternating Series Test to apply. Alternating series are usually flagged by (−1)ⁿ, (−1)ⁿ⁻¹, (−1)ⁿ⁺¹, or cos nπ (which works out to (−1)ⁿ). (However, if you have two of these expressions, the negatives cancel and the series is not alternating.) If other parts of the expression are not reliably positive (like sin n, which is sometimes positive and sometimes negative in no particular pattern), then AST does not apply.

• The AST is a one-way test − you can only use it to show that a series converges. If its conditions are not met, you cannot say that the series diverges. The AST simply tells you nothing, and you must find another test.

• You can only use the AST Error Estimate if the AST applies to the series.

• There are other Error Estimates from other tests (the Integral Test, for example), but the AST Error Estimate is the easiest to use if it applies.

• You can use the AST Error Estimate to determine the error when you add up a particular partial sum s_n , or you can use it to reverse-engineer what n should be in order to know that a partial sum s_n is accurate to within a specified error tolerance.