Series Expansions
Staff posted on October 23, 2006 |
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 {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


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

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