Series

9.2 SERIES

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

a₁ + a₂ + ... + a_n + ....

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

In n steps he will travel units,

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

It is natural to call this limit the infinite sum,

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 <a_n> we call it an infinite series and write it in the form

a₁ + a₂ + ... + a_n + ...

Given an infinite sequence <a_n>, each finite sum

a₁ + ... + a_n

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

a₁ + a₂ + ... + a_n+...,

there are associated two sequences, the sequence of terms,

a₁, a₂, ..., a_n,...,

and the sequence of partial sums,

S₁, S₂, ..., S_n, ...

where

S_n = a₁ + ... + a_n.

For each positive hyperreal number H, the infinite partial sum

S_H = a₁ + ... + a_H

is also defined, by the Extension Principle.