2 The radius of convergence
The most important statement one can make about a power series is that there exists a number, , called the radius of convergence, such that if the power series is absolutely convergent and if the power series is divergent. At the two points and the power series may be convergent or divergent.
Key Point 11
Convergence of Power Series
For a power series with radius of convergence then
the series converges absolutely if
the series diverges if
the series may be convergent or divergent at
For any particular power series the value of can be obtained using the ratio test. We know, from the ratio test that is absolutely convergent if
Example 2
-
Find the radius of convergence of the series
-
Investigate what happens at the end-points
of the region of
absolute convergence.
Solution
-
Here
so
In this case,
so the given series is absolutely convergent if and is divergent if .
-
At
the series is
which is divergent (the harmonic series). However, at
the series is
which is convergent (the alternating harmonic series).
Finally, therefore, the series
is convergent if
Task!
Find the range of values of for which the following power series converges:
First find the coefficient of :
Now find , the radius of convergence:
When the series is clearly divergent. Hence the series is convergent only if