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

[jdinatale]

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

 

[mathwonk]

"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"?

 

[jdinatale]

"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.

 

[JG89]

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"