Inside its range a power series behaves as though it were a polynomial: it may be differentiated and integrated term by term, and the resulting series has the same radius. That is a strong statement — differentiation was the operation uniform convergence failed to protect — and it is true here because the differentiated series is itself a power series whose radius can be computed and shown to be unchanged.

A machine that is assembled from infinitely many parts but behaves like a finite one inside a safe operating range. Outside the range the assembly falls apart, and inside it every operation that works on a finite assembly works.

Let $f(x)=\sum_{k=0}^{\infty}a_k(x-c)^k$ with radius $R>0$. Then $f$ is differentiable on $\lvert x-c\rvert<R$ with $f'(x)=\sum_{k=1}^{\infty}ka_k(x-c)^{k-1}$, and that series has radius $R$ as well. Integrating term by term is equally legitimate, and the antiderivative series also has radius $R$. Repeating gives $a_k=\tfrac{f^{(k)}(c)}{k!}$, so a power series is the Taylor series of its own sum.

• The radius is unchanged, although the behaviour at the endpoints may differ: $\sum\tfrac{x^k}{k^2}$ converges at both ends and its derivative series $$ at one.

The two upper limits differ by the factor contributed by the index, and that factor tends to one. Taking kth roots turns the factor k into the kth root of k, which converges to one, and multiplying a sequence by one that converges to one does not move its upper limit. So the reciprocal, which is the radius, is unchanged. Nothing about the coefficients themselves is used beyond that they are bounded in the relevant sense.

Take the kth root of the product.

The kth root of the index tends to one.

Multiplying by a sequence converging to one leaves an upper limit where it was.

The radius is the reciprocal of that upper limit, so it is unchanged.

Term by term differentiation and integration are both legitimate strictly inside the range.

Differentiating repeatedly and setting the variable to the centre recovers each coefficient.

• A power series may be differentiated and integrated term by term inside its range.
• The radius is unchanged; the endpoint behaviour may not be.
• $a_k=\tfrac{f^{(k)}(c)}{k!}$, so the representation is unique.

Great work! You now know how to:

• Differentiate and integrate a power series term by term
• Say why the radius is unchanged and what may change
• Read off the coefficients as Taylor coefficients
• Say why this does not contradict the derivatives lesson