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Series

9.2 SERIES

The sum of finitely many real numbers a1, a2, ..., an is again a real number a1 + a2 + ... + an. Sometimes we wish to form the sum of an infinite sequence of real numbers,

a1 + a2 + ... + an + ....

For example, if a man walks halfway across a room of unit width, then half of the remaining distance, then half the remaining distance again, and so forth, the total distance he will travel is an infinite sum

09_infinite_series-63.gif

In n steps he will travel 09_infinite_series-64.gif units,

09_infinite_series-65.gif

Thus he will get closer and closer to the other side of the room, and we have the limit

09_infinite_series-66.gif

It is natural to call this limit the infinite sum,

09_infinite_series-67.gif

We can go from this example to the general notion of an infinite sum. When we wish to find the sum of an infinite sequence <an> we call it an infinite series and write it in the form

a1 + a2 + ... + an + ...

Given an infinite sequence <an>, each finite sum

a1 + ... + an

is defined. This sum is called the nth partial sum of the series. Thus, with each infinite series

a1 + a2 + ... + an+...,

there are associated two sequences, the sequence of terms,

a1, a2, ..., an,...,

and the sequence of partial sums,

S1, S2, ..., Sn, ...

where

Sn = a1 + ... + an.

For each positive hyperreal number H, the infinite partial sum

SH = a1 + ... + aH

is also defined, by the Extension Principle.


Last Update: 2006-11-07