so
that 
. Note that
thisCalculus
Lesson
56
Obtaining
Power Series
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In this section, we will learn how to generate power series.
We will first use the geometric series.
We know so
that . Note that
this
is actually easier than most power series since all of the 
s are the same!
Also note that we already know the convergence properties of this series:
when 
, 
so the interval of
convergence is 
.
But what if we wanted it to be centered at 
so that we could
evaluate
the answer when x was near -1? Then we just use some algebra to make it
look like a geometric series centered at 
:
This is indeed a power series, centered at 
. Using the ratio test,
, thus 
.
Checking the endpoints, we find that the series diverges at both 
.
So the interval of convergence is 
.
The final answer, then, is that the power series, centered at 
, is
,
.
Another example: Centered at 
, find the power series for 
.
.
Example: Find the power series for 
centered at 
.
Example: Find the power series for 
centered at 
.
Separate by partial fractions and then add series together:
(radii calculated
separately or together).
Example: Find the power series for 
centered at 
.
We know
Note the change in interval of convergence from 1/x !!!!
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