Series of Functions
Let
Convergence
Convergence tests
Weierstrass M-test
To test if a series of functions converges uniformly and absolutely.
Let
converges
Then:
Differentiation
Theorem
If (all conditions must be met):
is differentiable ( is differentiable) on converges (pointwise) for some converges to uniformly on
Then:
converges to uniformly on is differentiable on OR in other words converges to uniformly
In that case, differentiation and infinite sum can be interchanged:
For power series
For any power series, inside the range of convergence, conditions for the above theorem is valid and thus the conclusions are valid.
Suppose