Series Expansions

In the fith century B.C. the theory on the limit of a sequence was introduced by the Greek philosopher Zeno of Elea.

By definition, a sequence {a_n} is a set of real numbers written in a define natural order.
For instance, the sequence {1, 1/2, 1/3, 1/4,…} can be described by a formula for n^th term
                  
{1, 1/2, 1/3, 1/4,…}  is called the range of the sequence

A sequence {a_n} has the limit L and is written, 

A series is formed by many terms (maybe infinitely many) added together. This is the basic

difference between series and sequences.

An infinite series(or simply a series) is denoted

The geometric series,

is convergent if | r | < 1 and its sum is

satifies
                 

Then, the series is convergent.

Alternating series estimation theorem