Real Analysis/Power Series
Please contribute to this section, it has been neglected. I have made a start, but it is not finished.
We have already encountered series. A power series is a series of the form
can also be seen as a real valued function. One of the first questions we will attempt to answer in this section is for which values of is is convergent.
We can use the root test here. We can see that
from before, so we see that there is a radius such that converges for and diverges for . This radius has a special significance in Complex analysis, but we will not be concerned with that here.
We can see by the root test that if is convergent, then is convergent, as , since , so this power series has the same radius of convergence as the original power series. Intuitively, we would guess that this was the derivative of it, but that requires proof. We look at the newton quotient:
One use of power series is to approximate functions. We can see that , so if a power series is a good approximation for , then .
We can also see from [[#Differentiability|]] that we need , and that , and by induction that , so we define the Taylor series of as
By translation, we can also approximate by
To do this, we of course require that is differentiable an infinite number of times at o or t.