When given a power series, we know that it converges on some neighborhood of its center (perhaps including some endpoints of this neighborhood). Its convergence can be therefore described by providing these basic parameters: The center, the radius of convergence, the region of absolute convergence, and the region of convergence. Sometimes we are not all that much interested in what happens at endpoints, then the first two pieces of data (center and radius) are considered sufficient.
1. Determining center. This is actually very simple. The "official"
form of a power series features terms
2. Determining radius of convergence. This is done by investigating
absolute convergence of the given series.
Step 1. Apply absolute value to terms of the series and simplify, in
particular you should get the term
Step 2. Apply either the
Root test
or the
Ratio test
to this series. The corresponding constant in most cases (but not
necessarily) features
Step 3. Check on convergence by solving
If the term
Alternative: Some people prefer to use the formula
Its disadvantage is that it can be used only for series in the "official" form, whereas the procedure described above can be also applied to series in different forms (see Example below).
3. Determining behavior at endpoints. Endpoints are
Note that this only applies to the case of positive R, there are no
endpoints when R is zero or infinity.
4. Regions of convergence. If the radius of convergence is positive,
then the region of convergence is the interval given by the center and the
radius,
Usually the open interval above is also the region of absolute convergence. The
only exception is the case when we get convergence at both endpoints, in
which case the region of absolute convergence is usually the closed interval as
above and it coincides with the region of convergence.
If
If
Remark: Often we are given a power series that is not in its proper
form, which usually means that instead of
Example: Investigate convergence of the series
First we rearrange the series so that it has the proper form.
Thus the center is
Thus we get the radius of convergence
From the center and radius of convergence we see that endpoints are −4 and 0 (see picture below), now we check on them.
x = −4: We substitute into the given series (as we rewrote it above) and obtain
This is the famous alternating harmonic series that is known to converge by the Alternating series test, see the example there.
x = 0: We substitute into the given series and obtain
This is the equally famous harmonic series that is known to diverge (or we apply the p-test).
Conclusion: The region of convergence is the interval
Some people find it easier to work with these notions if they draw a picture and fill in data at every step.
Remark: The series in the above example was not given in its proper form, how would we apply the above algorithm to this series as given?
The center is found by solving the equation
The constant ϱ is found similarly as above.
This way it is usually easier. However, if you are used to determine the radius of convergence using that formula, then you necessarily do have to change the series to its standard form.
For other examples see Solved problems - Series of functions.
Expanding in power series
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of functions