Real Analysis/Power Series

Please contribute to this section, it has been neglected. I have made a start, but it is not finished.

Definition edit

We have already encountered series. A power series is a series of the form

  where  .

  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.

Convergence edit

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.

Differentiability edit

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:





Taylor Series edit

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.