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# Alternating Series for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

Main Concepts: Alternating Series, Error Bound, Absolute Convergence

### Alternating Series

A series whose terms alternate between positive and negative are called alternating series. Alternating series have one of two forms: An alternating series converges if a1 > a2 > a3 > … > an > … and

#### Example 1

Determine whether the series converges or diverges.

Step 1:

Step 2: Note that and in general, since multiplying by en+1 gives en > n+1.

Step 3: ≈ {.36788, .27067, .14936, .07326, …}, so Therefore the series converges.

#### Example 2

Determine whether the series converges or diverges. If it converges, find its sum.

Step 1: is a geometric series with a = 4 and Since r < 1, the series converges.

Step 2:

### Error Bound

If an alternating series converges to the sum S, then S lies between two consecutive partial sums of the series. If S is approximated by a partial sum sn, the absolute error |Ssn| is less than the next term of the series an +1, and the sign of Ssn is the same as the coefficient of an +1.

#### Example

converges to 3.2. This value is greater than sn for n odd, and less than sn for n even. If S is approximated by the third partial sum, s3 =3.25, the absolute error |Ss3|=|3.2 – 3.25|=| – 0.05|=0.05, which is clearly less than The coefficient of a4 is negative, as is Ss3.

### Absolute Convergence

If the series converges, then converges. A series that converges absolutely converges.

#### Example

Determine whether the series converges.

Step 1:   Consider the series . For this series, s1 = 1, , , . The sequence of partial sums, , converges to 1.5. Or, note that this is a geometric series with a = 1, r = ; thus it converges to .

Practice problems for these concepts can be found at: Series Practice Problems for AP Calculus

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