Sequence and Series Dr. Geetha Sivaraman Department of Mathematics St.Joseph’s College(Autonomous), Tiruchirappalli Geetha

Definition of the Limit of a Sequence Geetha

Limit of a Sequence Geetha

Properties of Limits of Sequences Geetha

Squeeze Theorem for Sequences Geetha

Absolute Value Theorem Geetha

Definition of a Monotonic Sequence Geetha

Definition of a Bounded Sequence Geetha

Bounded Monotonic Sequences Geetha

Problems: 1. Prove that convergence of implies the convergence of Problems: 1. Prove that convergence of implies the convergence of . Check whether the converse is true. 2. calculate 3. Discuss the convergence of by assuming Geetha

Ans: 1) Geetha

Ans 2) ½ Ans 3) Geetha

Definitions of Convergent and Divergent Series Geetha

Convergence of a Geometric Series Positive term series : A series in which the terms are positive after some stage N. Geetha

Properties of Infinite Series The convergence or divergence of a positve term series will not be affected if i).by multiplying a non zero scalar to each term ii). By adding or deleting some finite terms. Geetha

The Integral Test Geetha

Convergence of p-Series Geetha

Limit of nth Term of a Convergent Series Geetha

nth-Term Test for Divergence Geetha

Direct Comparison Test Geetha

Limit Comparison Test Geetha

Problems:   Geetha

Determine the Nature of the following Series   Geetha

Comparision of Ratios: Geetha

D’ Alembert’s Ratio Test Geetha

Ratio Test Geetha

Raabe’s Test Geetha

Exercise:   Geetha

Answers:   Geetha

Answers:   Geetha

Answer   Geetha

Answer   Geetha

Test the convergence of the following Series   Geetha

Answers   Geetha

Cauchy’s Root Test Geetha

  Geetha

Guidelines for Testing a Series for Convergence or Divergence Geetha

Summary of Tests for Series Geetha

Summary of Tests for Series (cont’d) Geetha

Alternating Series Test Geetha

Alternating Series Remainder Geetha

Absolute Convergence Geetha

Definitions of Absolute and Conditional Convergence Geetha

Definitions of nth Taylor Polynomial and nth Maclaurin Polynomial Geetha

Taylor's Theorem Geetha

Definition of Power Series Geetha

Convergence of a Power Series Geetha

Properties of Functions Defined by Power Series Geetha

Operations with Power Series Geetha

The Form of a Convergent Power Series Geetha

Definitions of Taylor and Maclaurin Series Geetha

Convergence of Taylor Series Geetha

Guidelines for Finding a Taylor Series Geetha

Power Series for Elementary Functions Geetha