Geometric+Sequences+and+Series,+notation+and+geometric+sums

=__** Geometric Series, Notation and Geometric sums **__=

A geometric sum has a general form where a is the first term in the series and r is the common ratio.

This sum can be written in sigma notation like this:

Thus series can be rewritten to fit the general form and then expanded as a power series.

When rewritten to fit the general form of a geometric sum it looks like this (series is centered at x = 0):



Next this can be expanded to be:

Graphically, this would look like:



A geometric series is convergent for |r|<1, so for this series we can say that the interval for convergence is |-2x|<1. Simplified, the interval of convergence is -1/2<x<1/2. Looking at the graph, the series is convergent from the asymptote at x= -1/2 to the value x=1/2. For this series it is important to recognize that it is centered at x=0, but this does not necessarily have to be the case. If we wanted to center this at x=1, we substitute (x-1) for x. The series would then become. You have to add the two on the bottom because you have to account for the two that is subtracted within the parenthesis. This can then be simplified to. To fit the general form we could manipulate this expression to be.

This can expressed graphically like this:

As you might notice, this is the exact same graph as the original function because it models the only thing that is changed is where the series is centered. Now we can find the interval of convergence by |-2/3(x-1)|<1. -1<2/3(x-1)<1 -3/2<x-1<3/2 -1/2<x<5/2 The series is thus convergent on this interval. The only difference between this power series and the original power series is the point at which it is centered.

Through doing this wiki, we found that for this particular function, for all values from 0 to infinity that the function is centered on, it is always convergent from the asymptote at x=-.5 to some value( two times the center point minus the asymptote). As the center point for the series is moved further to the right, the greater the interval convergence becomes. If the function was centered at infinity, it is hypothesized that it would be the exact value of the function (1/(1+2x)).
 * "Fun Fact"**

To summarize, often rational functions can be manipulated to fit the form, where they can be easily transformed into an infinite series. This infinite series can be approximated by a Taylor series. By doing this, you can further manipulate the function.

http://www.wolframalpha.com/input/?i=series+expand%3A+{1%2F(1%2B2x)} http://www.wolframalpha.com/input/?i=series+expand%3A+{1%2F(3%2B2(x-1))} http://www.wolframalpha.com/input/?i=series+expand%3A+{1%2F(1%2B2x)}