Ch 9.4 Radius of Convergence Calculus Graphical, Numerical, Algebraic by Finney, Demana, Waits, Kennedy.

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Presentation transcript:

Ch 9.4 Radius of Convergence Calculus Graphical, Numerical, Algebraic by Finney, Demana, Waits, Kennedy

Importance of Convergence

This equation is an identity only for x in (-1,1). Convergence matters!

What You’ll Learn When does a power series converge? If a power series converges, over what interval of x? –The n th Term Test for Divergence There are 4 tests for convergence: –The Comparison Test –Absolute Convergence Test for Alternating Series –The Ratio Test –Endpoint Convergence

Note: R is the radius of convergence. This result is useless since the series is only accurate at the center.

Cases of Convergence Case 1: Converges over a radius of 1 or -1 < x < 1. Case 2: Converges for all values of x. Case 3: Converges only at center or x = a. We will see an example for homework.

The nth Term Test for Divergence for non-negative terms only First we need to rule out divergence:

Direct Comparison Test

Example of Comparison Test

Note: This test is for alternating series.

Example of Absolute Convergence

If L = 1, Ratio Test is Inconclusive

Endpoint Convergence

Summing a Telescoping Series