Monday, February 11 −Lecture 18: Tests for convergence of series : Comparison test.(Refers to Section 8.4 in your text) After having practiced the problems associated to the concepts of this lecture the student should be able to: Apply the comparison test to determine convergence or divergence of a series, show a reasonable error bound for the value of a series known to converge by the integral test or by the fact it is a geometric series. 18.1Theorem −The Comparison test. Let be two series such that 0 ≤ai ≤ bi for all i. Then the following statements hold true : Proof: Part I : In this part the Monotone convergence (sequence) theorem is invoked. Given: Since 0 ≤ai ≤ bi for all i. So the sequences {An} and {Bn} are such that An ≤ Bn for all n. We are also given that the sequence {Bn} converges, say to L. Required to show: That the sequence {An} must converge to some number.

Part II : Given: The sequences {An} and {Bn} are such that An ≤ Bn for all nand that the sequence {An} diverges.