CHAPTER Continuity Series Definition: Given a series   n=1 a n = a 1 + a 2 + a 3 + …, let s n denote its nth partial sum: s n =  n i=1 a i = a 1 + a 2 + … + a n. If the sequence {s n }is convergent and lim n   s n = s exists as a real number, then the series  a n is called convergent and we write a 1 + a 2 + a n + …= a i or   i=1 a n = s. The number s is called the sum of the series. Otherwise, the series is called divergent.

CHAPTER Continuity The geometric series   n=1 ar n-1 = a + ar + ar 2 + … Is convergent if | r | < 1 and its sum is   n=1 ar n-1 = a / (1–r) | r | < 1. If | r |  1, the geometric series is divergent. Theorem: If the series   n=1 a n is convergent, then lim n   a n = 0. The Test for Divergence: If lim n   a n does not exist or lim n   a n  0, then the series   n=1 a n is divergent.

CHAPTER Continuity Theorem: If  a n and  b n are convergent series, then so are the series  ca n (where c is a constant),  (a n + b n ), and   n=1 (a n - b n ), and (i)   n=1 ca n = c   n=1 a n (ii)   n=1 (a n + b n ) =   n=1 a n +   n=1 b n (iii)   n=1 (a n - b n ) =   n=1 a n -   n=1 b n.