1549901369-Elements_of_Real_Analysis__Denlinger_

454 Chapter 8 • Infinite Series of Real Numbers

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If L an = +oo (or -oo) we do not say that I.: an converges, but rather say
n=l n=l
that it diverges to +oo (or -oo).
We occasionally forego the sigma notation, and write the series (1) as

In this notation,

and in case of convergence,

a1 + a2 + a3 +···+an+··· = n-+oo lim (a1 + a2 + a3 +···+an)·

Thus, it makes sense to write

oo n

Lan= lim L ak

n=l n-+oo k=l

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whenever this limit exists. The symbol I.: an can be used to represent either

n=l

the series or its sum; we depend on the context to make clear which we mean.

As with sequences, we may begin the subscripts in a series with an integer

other than 1. Thus, we may consider series of the form

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L an = an 0 + ano+l + an 0 +2 + · · ·

n=no

as long as each term an is defined for all n 2:: no. Furthermore, when it will not

lead to confusion, we often denote a series merely by the more generic symbol,

L.:an.

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Example 8.1.2 The geometric series I.: arn =a+ ar + ar^2 + ar^3 + · · ·

n=O

In Exercise 1.3.12 we saw that the nth partial sum of this series is

n a - arn+l

Sn = I.: ark = ----

k=O 1 - r

a

= -[1-rn+I]'

1-r

and in Theorem 2.4.6 we noted that lim rn

n-+oo {

0 if Jrl < 1}

1 if r = 1. Thus, if

+oo if r > 1.