Recall that the geometric series
is convergent exactly when -1<q<1.
when -1<x<1. We can rewrite the function as a series! Consider another example: What about rewriting ? Rewrite:
and use the formula for the geometric series with :
Since the geometric series formula "works" for |q|<1, this series expansion will work exactly when , i.e., when |x|<1. (Check this carefully!!!)
If we plug in x=0 on both sides, using , we obtain C=0 and thus
Let's check graphically whether this might work: The graph of 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
as an infinite sum of the product of certain numbers and powers of the variable x. Such expressions are called power series with center 0; the numbers are called its coefficients. Slightly more general, an expression of the form
is called a power series with center .
Using the summation symbol we can write this as
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