Introduction to Real Analysis

Dr. Weihu Hong

Clayton State University

10/7/2008

Series of Real Numbers

Definition 2.7.1 Let be a sequence in R, and let be the sequence obtained from , where for each nєN, .The sequence is called an infinite series, or series, and is denoted either as For every nєN, is called the nth partial sum of the series and is called the nth term of the series.

n

kkn ps

1

1}{ nnp

nssps

divergesconvergessdivergesconvergesp

nk

k

nnk

k

lim

/}{/

1

11

1}{ nns

1}{ nnp

1}{ nns

.211

n

kk ppporp

ns

np

Examples

(a) Geometric series

(b) Consider the series .

(c) Consider the series

1k

kr

1 )1(

1

k kk

1

1

kpk

Theorem 2.7.3 (Cauchy Criterion)

The series converges if and only if given ε>0,

there exists a positive integer K such that

1kkp

Knmallforpm

nkk

1

Corollary 2.7.5

If converges, then

Remark. Is the following statement true?

If , then converges.

1kkp 0lim

kk

p

1kkp0lim

kk

p

Theorem 2.7.6

Suppose for all nєN. Then

Why the above theorem doesn’t apply to the series

.}{1

aboveboundedissconvergesp nk

k

0np

1

1)1(k

k

Structure of Point Sets

Definition 3.1.1 Let E be a subset of R. A point pєE 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 3.1.3 (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.

EpN )(

FRF c \