Power Series
Definition of a Power Series
Definition of a Power Series
Let f(x) be the function represented by the series
^{
}
Then f(x) is called a power series
function. 
More generally, if f(x) is represented by the series
^{
}Then we call f(x) a power series centered at x =
c. The domain of f(x) is called the Interval of Convergence and half
the length of the domain is called the Radius
of Convergence.
The Radius of Convergence
To compute the radius of convergence, we use the ratio test.
Example: Find the radius of convergence of
^{
}
Solution: We use the Ratio Test:
We solve
or x  3 < 2
so that
1 < x < 5
Since
1
(5  1) = 2
2
the radius of convergence is 2. Notice that
we could have use the geometric series test and obtained the same result.
The ratio test is the most likely test to work, but occasionally another test
such as the geometric series test or the root test is easier to use.
Exercise:
Find the radius of convergence of
^{
}
Interval of Convergence
To find the interval of convergence we follow the three steps:

Use the ratio test to find the interval where the series is absolutely
convergent.

Plug in the left endpoint to see if it converges at the left endpoint.
(AST may be useful).

Plug in the right endpoint to see if it converges at the right endpoint.
(AST may be useful).
Example:
Find the interval of convergence for the previous example:
Solution:

We have already done this step and found that the series converges
absolutely
for 1 < x < 5.
We plug in x = 1 to get
^{
}
This series diverges by the limit test.

We plug in x = 5 to get
This series also diverges by the limit test.
Hence the endpoints are not included in the interval of convergence. We
can conclude that the interval of convergence is
1 < x < 5
Exercise Find the interval of convergence of the previous exercise:
^{
}
Differentiation and Integration of Power Series
Since a power series is a function, it is natural to ask if the function is
continuous, differentiable or integrable. The following theorem
answers this question.
Theorem
Suppose that a function is given by the power series
^{
}and that the interval of convergence is
(c 
R, c + R) (plus possible endpoints)
then f(x) is continuous, differentiable, and integrable on that interval
(not necessarily including the endpoints). To obtain the derivative or the integral of
f(x) we can pass the derivative or integral through the
S. In other
words
and
Furthermore, the radius of convergence for the derivative and integral is R.

Example:
Consider the series
^{
}by the GST this series converges for x <
1, hence the center of convergence is 0 and the radius is
1.
By the above theorem
^{
}has center of convergence 0 and radius of convergence
1 also. We can also say that
also has center of convergence 0 and radius of convergence
1.
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