Power Series


Recall that the geometric series

displaymath88

is convergent exactly when -1<q<1.


Rename "q" to "x", and flip sides:

displaymath89

when -1<x<1. We can rewrite the function tex2html_wrap_inline112 as a series! Consider another example: What about rewriting tex2html_wrap_inline114 ? Rewrite:

displaymath90

and use the formula for the geometric series with tex2html_wrap_inline116 :

eqnarray22

Since the geometric series formula "works" for |q|<1, this series expansion will work exactly when tex2html_wrap_inline120 , i.e., when |x|<1. (Check this carefully!!!)


Let's dream on, and integrate both sides: tex2html_wrap_inline124 , so we obtain:

displaymath91

If we plug in x=0 on both sides, using tex2html_wrap_inline128 , we obtain C=0 and thus

displaymath92

Let's check graphically whether this might work: The graph of tex2html_wrap_inline132 is black, the sum of the first terms on the right are depicted in red. (The "number of terms" in the picture actually also counts the terms with zero coefficients!)

It seems to work as long as -1<x<1. With a little bit of work, the formula for the geometric series has led to a series expression for the inverse tangent function!

As it turns out, many familiar (and unfamiliar) functions can be written in the form

displaymath93

as an infinite sum of the product of certain numbers tex2html_wrap_inline136 and powers of the variable x. Such expressions are called power series with center 0; the numbers tex2html_wrap_inline136 are called its coefficients. Slightly more general, an expression of the form

displaymath94

is called a power series with center tex2html_wrap_inline142 .

Using the summation symbol we can write this as

displaymath95


Try it yourself!

Use the fact that tex2html_wrap_inline152 to write down a power series representation of the logarithmic function tex2html_wrap_inline154 . What is the center of the power series? For what values of x will this representation be valid? You might want to check your answer graphically, if you have a graphing calculator or access to a Math software program.

Click here to see the answers and to continue.


Helmut Knaust
Wed Jul 10 13:02:07 MDT 1996

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