When do we have (lim f_n )' = lim f'_n?

Question

Suppose the sequence of real valued function f_n converges to some f on an interval [a,b] and suppose each f_n is differentiable. I know this doesn't automatically implies f'_n converges to f'. Actually, f'_n doesn't need to converge and, even if it does, it's limit may be other than f'. Is there any theorem which ensures that, under some suitable assumptions, we actually have f' = (lim f_n )' = lim f'_n? Could it's proof be give n here or is it too long for YA?

I know the desired condition is always satisfied by power series. Why? Why are power series so nice, why do they behave so well?

Thank you for your time.

Answer 1

Actually, there's a theorem that ensures the desired result:

Let (f_n) be a sequence of real valued functions defined and differentiable on a finite interval [a,b] (or (a, b), this doesn't matter). Suppose that, for some x_0 in [a,b], the sequence of real numbers (f_n(x_0)) is convergent and that the sequence (f'_n) converges UNIFORMLY on [a, b] to a function g. Then, (f_n) converges uniformly on [a, b] to a differentiable function f such that f'(x) = g(x) for every x in [a, b].

So, the important thing here is that the sequence of DERIVATIVES converges UNIIFORMLY. If this is satisfied, then it's enough that f_n converges at a single point in order for the theorem to be true.

Like you said, convergence of the primitives f_n, even if uniform, implies absolutely nothing about the sequence f'_n.

The proof of this theorem is not difficult, but is actually a bit long. I suggest you consult a good book on Real Analysis, like those I mention in References. If I presented the proof here I'd just repeat the proofs you can find in the books, I wouldn't add anything new. I've seen 3 proofs for this theorem. Though they are not exactly the same, they follow the same idea. If you understand one of them, you automatically understood the other 2.

Power series are really nice. Functions given by power series are called analytic, and they are best studied in the domain of Complex Analysis. The have derivatives of all orders. In te real case, it's possible to prove the result you mentioned invoking the theorem I gave here. But this is not the best approach. The natural domain to study power series is really complex analysis.

I think one of the reasons that make power series nice is that all of their derivatives converge in the interior of the same disk of convergence and converge uniformly on every compact subset of such circle.