Library Articles


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 {an} 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 nth term
                  
{1, 1/2, 1/3, 1/4,...}  is called the range of the sequence

A sequence {an} 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


The geometric series,

                

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

                


The Alternating Series




satifies
                 

Then, the series is convergent.

Alternating series estimation theorem



                 


The Root and Ratio Tests

The Ratio Test




The Root Test

 

Power Series

Taylor and Maclaurin Series











Important Maclaurin series and there intervals of convergence:

                        
                        
                        
                        
                        

The Binomial Series