# Is showing that a series of functions is differentiable as simple as it appears?

Here is the problem and my want. I think I might be overlooking something because it seems rather simple...

mathwonk
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"Hence"? how do you know you can differentiate this infinite sum of differentiable functions? what rule tells you that? do you know the weierstrass "M test"?

"Hence"? how do you know you can differentiate this infinite sum of differentiable functions? what rule tells you that? do you know the weierstrass "M test"?

Yes, I have access to the Weierstrass M-test. But that seems to deal with uniform convergence, and it doesn't quite seem to deal with differentiability or continuity. Unless I'm misunderstanding it.

Straight from a book:

"If, on differentiating a convergent infinite series $\sum_{v = 0}^{\infty} G_v(x) = F(x)$ term by term, we obtain a uniformly convergent series of continuous terms $\sum_{v = 0}^{\infty} g_v(x) = f(x)$, then the sum of this last series is equal to the derivative of the sum of the first series"