What does tbis thoerem represent ??? what it wants to say

Tences (Real Ar al Analysis Cal SCen Tences Theorem : Let [a, b] be a closed and bounded interval and for each n e N, let f, be differentiable on a, b]. If each f, be continuous on [a, b] and the series of functions f, + f2+ f, t.. converges uniformly on [a, b] to a function g and the series f, + f, + f, + .. converges to s on [a, b], then s'(x) = g(x) for all x e [a, b]. 9PR 3 la Proof : Let s x) = f(x) + f.(x) + + f (x)