Convergence of Power Series
We’ll just consider our power series to be in , because if we have a real power series we can consider each coefficient as a complex number instead. Now we take a complex number and try to evaluate the power series at this point. We get a series of complex numbers
by evaluating each polynomial truncation of the power series at and taking the limit of the sequence. For some this series may converge and for others it may not. The amazing fact is, though, that we can always draw a circle in the complex plane — — within which the series always converges absolutely, and outside of which it always diverges. We’ll say nothing in general about whether it converges on the circle, though.
The tool here is the root test. We take the th root of the size of the th term in the series to find . Then we can pull the -dependance completely out of the limit superior to write . The root test tells us that if this is less than the series will converge absolutely, while if it’s greater than the series will diverge.
So let’s define . The root test now says that if we have absolute convergence, while if the series diverges. Thus is the radius of convergence that we seek.
Now there are examples of series with all sorts of behavior on the boundary of this disk. The series has radius of convergence (as we can tell from the above procedure), but it doesn’t converge anywhere on the boundary circle. On the other hand, the series has the same radius of convergence, but it converges everywhere on the boundary circle. And, just to be perverse, the series has the same radius of convergence but converges everywhere on the boundary but the single point .