Introduction to Real Analysis Dr. Weihu Hong Clayton State University 10/7/2008

Series of Real Numbers Definition Let be a sequence in R, and let be the sequence obtained from, where for each n єN,.The sequence is called an infinite series, or series, and is denoted either as For every n єN, is called the nth partial sum of the series and is called the nth term of the series.

Examples (a) Geometric series (b) Consider the series. (c) Consider the series

Theorem (Cauchy Criterion) The series converges if and only if given ε>0, there exists a positive integer K such that

Corollary If converges, then Remark. Is the following statement true? If, then converges.

Theorem Suppose for all n єN. Then Why the above theorem doesn’t apply to the series

Structure of Point Sets Definition Let E be a subset of R. A point p єE is called an interior point of E if there exists an ε>0 such that The set of interior points of E is denoted by Int(E). Definition (a) A subset O of R is open if every point of O is an interior point of O. (b) A subset F of R is closed if is open.