Does the Series Converge? 10 Tests for Convergence nth Term Divergence Test Geometric Series Telescoping Series Integral Test p-Series Test Direct Comparison Test Limit Comparison Test Alternating Series Test/Absolute Convergence Test Ratio Test Root Test Each test has it limitations (i.e. conditions where the test fails). The test tells you nothing!

11.5A Alternating Series – terms alternate in signs OR NOTE: All a n ’s are assumed to be positive.

OTHER FORMS OF ALTERNATING SERIES Instead of using to create an alternating series, can be used. Be careful… a series with both positive and negative terms is not alternating unless every other term alternates between positive and negative, with the absolute value of all terms being generated with the same rule for.

Alternating Series - Examples The alternating harmonic series (will prove to be convergent). An alternating geometric series (convergent because r = –1/2). A divergent alternating series (nth-term test).

The Alternating Series Test The series … Converges if …

The Alternating Series Test Converges if … “Proof”:

Alternating Series - Examples

11.5B Approximating Alternating Series If an alternating series satisfies the conditions of the alternating series test, and S N, the partial sum of the first N terms, is used to approximate the sum, S; then … The error, R N, is less than the first term omitted.

Approximating Alternating Series Example: 1. Determine the sum of the first 4 terms.

Approximating Alternating Series Example: 2.Estimate the error if 4 terms are used to approximate the sum. 3.Therefore the sum, S, lies between:

Approximating Alternating Series Example: 2. How many terms are needed to make sure the error is less than 0.01? Therefore, four terms are needed!

Example 1 of the Alternating Series Test Decreasing? Limit? Therefore, convergent. The Alternating Harmonic Series

Example 2 of the Alternating Series Test Decreasing? Limit? Therefore, convergent.

Absolute Convergence converges absolutely … converges. if converges conditionally … if converges but... diverges.

Absolute Convergence: Example 1 Divergent harmonic series, therefore the alternating series is conditionally convergent but not absolutely convergent. is a convergent alternating series.

Absolute Convergence: Example 2 Convergent p-series, therefore the alternating series is absolutely convergent. is a convergent alternating series.

Absolute Convergence: Example 3 Convergent geometric series, therefore the first series converges absolutely. Therefore, the original series converges.

Absolute Convergence Test: Ex. 1 Convergent geometric series, therefore the first series converges absolutely. If a series converges absolutely, it is a convergent series. Note that the first series is NOT an alternating series.