Chapter 3 - CBU

3.7. INTRODUCTION TO INFINITE SERIES 1453.7. Introduction to Infinite SeriesDefinition (3.7.1). If X = (x n ) is a sequence in R, then the infinite series(or just series) generated by X is the sequence S = (s k ) defined bys 1 = x 1s 2 = s 1 + x 2 (= x 1 + x 2 ).s k = s k 1 + x k (= x 1 + x 2 + · · · + x k ).The x n are the terms of the series and the s k are the partial sums of the series.If lim S exists, we say the series is convergent and call this limit the sum orvalue of the series. If this limit does not exist, we say this series S is divergent.Notation.We can also useX(xn )1Xn=0orx nXxnIf the first term of the series is x N , then the first partial sum is s N .or1Xn=5orx n1Xn=1x n