Prove the following:
Let  be a sequence of functions continuous on a set 
containing the contour , and suppose that converges uniformly to on .

Then the series converges to .

Using this result and the Generalized Cauchy Integral formula for derivatives (see below),

show the following:
If all  are analytic throughout a domain D and converges uniformly to 
on any closed subdisk of , then the derivative exists and converges to 

for every .

The Generalized Cauchy Integral formula for derivatives (no need to prove):

If is analytic inside and on the simple closed positively oriented contour 
and if is any point inside , then

,	n = 1,2,3,...