## Derivatives of Power Series

The uniform convergence of a power series establishes that the function it represents must be continuous. Not only that, but it turns out that the limiting function must be differentiable.

A side note here: we define the derivative of a complex function by exactly the same limit of a difference quotient as before. There’s a lot to be said about derivatives of complex functions, but we’ll set the rest aside until later.

Now, to be specific: if the power series converges for to a function , then has a derivative , which itself has a power series expansion

which converges within the same radius .

Given a point within of , we can expand as a power series about :

convergent within some radius of . Then for in this smaller disk of convergence we have

by manipulations we know to work for series. Then the series on the right must converge to a continuous function, and continuity tells us that each term vanishes as approaches . Thus exists and equals . But our formula for tells us

Finally, we can apply the root test again. The terms are now . Since the first radical expression goes to , the limit superior is the same as in the original series for : . Thus the derived series has the same radius of convergence.

Notice now that we can apply the exact same reasoning to , and find that *it* has a derivative , which has a power series expansion

which again converges within the same radius. And so on, we determine that the limiting function of the power series has derivatives of arbitrarily large orders.

How did you obtain the equation of:

?

Since about , the above becomes

,

which is different from your result.

Comment by Bernd | September 23, 2008 |

Where do you get that ? I just gave the power series expansion about , which clearly indicates that .

Comment by John Armstrong | September 23, 2008 |

[...] Okay, we know that power series define functions, and that the functions so defined have derivatives, which have power series expansions. And thus these derivatives have derivatives themselves, and so [...]

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