Series and Convergence Lesson 9.2

Definition of Series Consider summing the terms of an infinite sequence We often look at a partial sum of n terms

Definition of Series We can also look at a sequence of partial sums { S n } The series can converge with sum S The sequence of partial sums converges If the sequence { S n } does not converge, the series diverges and has no sum

Examples Convergent Divergent

Telescoping Series Consider the series Note how these could be regrouped and the end result As n gets large, the series = 1

Geometric Series Definition An infinite series The ratio of successive terms in the series is a constant Example What is r ?

Properties of Infinite Series Linearity The series of a sum = the sum of the series Use the property

Geometric Series Theorem Given geometric series (with a ≠ 0) Series will Diverge when | r | ≥ 1 Converge when | r | < 1 Examples Compound interest Or

Applications A pendulum is released through an arc of length 20 cm from vertical Allowed to swing freely until stop, each swing 90% as far as preceding How far will it travel until it comes to rest? 20 cm

Assignment Lesson 9.2 Page 612 Exercises 1 – 69 EOO