POWER SERIES AND THE USES OF POWER SERIES
Elizabeth Wood
Now we are finally going to start working with a topic that uses all of the information
from the previous topics. The topic that we are going to discuss is the power series and
the use of this series. First of all, let us define what a power series.
FACT:
A power series about x = a is the series of the form
The big question is when will a power series converge absolutely, converge
conditionally, or diverge. To do this we will use a modified ratio test, and it will look
like the following.
Then you will have to determine where this expression is less than one.
R is called the radius of convergence, and a - R < x < a + R is the interval of
convergence. The power series converges absolutely for any x in that interval.
Then we will have to test the endpoints of the interval to see if the power series might
converge there too. If the series converges at an endpoint, we can say that it converges
conditionally at that point. Any value outside this interval will cause the power series
to diverge.
EXAMPLE 1:
Find the following series' radius and interval of convergence. For
what values of x does the series converge absolutely, or
conditionally.
SOLUTION:
First apply the modified ratio test remembering that x is a constant.
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The radius of convergence for this series is 1. Now to determine
the interval of convergence.
| x + 5 | < 1  -1 < x + 5 < 1  -6 < x < -4
The interval of convergence is -6 < x < -4. This is where the series
will converge absolutely. Now we must test the endpoints.
This is a geometric series with | r |  1, therefore it diverges.
This is a geometric series with | r |  1, therefore it diverges.
Therefore, there are no values for which this series converges
conditionally.
EXAMPLE 2:
Find the following series' radius and interval of convergence. For
what values of x does the series converge absolutely, or
conditionally.
SOLUTION:
Again apply the modified ratio test and hold x constant.
The radius of convergence for this series is 1. Now to determine
the interval of convergence.
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| x + 2 | < 1  -1 < x + 2 < 1  -3 < x < -1
Therefore, the interval convergence is -3 < x < -1. This is where
this series will converge absolutely. Now we must test the
endpoints.
This is the harmonic series, and this series diverges.
This is the alternating harmonic series, and this series converges.
Therefore, this series converges conditionally at x = -1.
EXAMPLE 3:
Find the following series' radius and interval of convergence. For
what values of x does the series converge absolutely, or
conditionally.
SOLUTION:
Remember that x is a constant, so as n goes to infinity, the limit is
zero. So this series converges absolutely for all x. The radius of
convergence is  and the interval of convergence is (- , ).
EXAMPLE 4:
Find the following series' radius and interval of convergence. For
what values of x does the series converge absolutely, or
conditionally.
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SOLUTION:
Since the limit is greater than 1, this series will only converge
absolutely for x = 0. The radius of convergence for this series is 0.
All other values of x will cause this power series to diverge.
1.
There is a positive number R such that the series diverges for | x - a | > R, but
converges absolutely for | x - a | < R. The series may or may not converge at
either endpoints x = a - R and x = a + R.
2.
The series converges absolutely for all x (R =  ).
3.
The series converges absolutely at x = a and diverges everywhere else (R = 0).
EXAMPLE 5:
Find the following series' radius and interval of convergence. For
what values of x does the series converge absolutely, or
conditionally.
SOLUTION:
Therefore, the radius of convergence is 4/3. Now to determine the
interval of convergence.
The interval of convergence is
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and, this is where the series will converge absolutely. Now we
must test the endpoints.
This is a geometric series with | r |  1, therefore it diverges.
This is a geometric series with | r |  1, therefore it diverges. There
are no values for which this power series converges conditionally.
EXAMPLE 6:
Find the following series' radius and interval of convergence. For
what values of x does the series converge absolutely, or
conditionally.
SOLUTION:
The radius of convergence is 2.
| x - 3 | < 2  -2 < x - 3 < 2  1 < x < 5
The interval of convergence is 1 < x < 5, and this is the interval
where this series converges absolutely. Now to test the endpoints.
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This is a geometric series with ratio | r |  1, therefore it diverges.
This is a geometric series with | r |  1, therefore it diverges. There
are no values for which this power series converges conditionally.
At the beginning of this set of supplemental notes, I stated that there are uses for the power
series, now is the time to start discussing them. The main use of a power series is to numerically
approximate integrals or derivative of functions that are not easily integrated or differentiated.
We can do term-by-term differentiation of a power series of a function as long as x is inside the
interval of convergence. This is also true for term-by-term integration.
Consider the power series
with | x | < 1. This is a geometric series with ratio less than one, so it sum is
Now, let us integrate both sides of this equation.
Let u = 1 - x, then du = -dx.
Suppose, I wanted to find the derivative of
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We will discuss this topic more after we discuss Taylor series. Power series will be very
important in differential equations and other engineering courses, so I would make sure that I
understand the topics that are associated with them. Work through these examples making sure
to understand all of the steps.
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