Series of Functions

Let is a sequence of integrable functions. And series of those functions is defined as:

Convergence tests

Weierstrass M-test

To test if a series of functions converges uniformly and absolutely.

Suppose is a sequence of functions on a set . If both these conditions are met:

• converges

Then:

Proof Hint

• is Cauchy
• is uniformly Cauchy which implies it’s converging uniformly and absolutely

Differentiation

Theorem

If (all conditions must be met):

1. is differentiable ( is differentiable) on
2. converges (pointwise) for some
3. converges to uniformly on

Then:

1. converges to uniformly on
2. is differentiable on
3. 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 , and is the radius of convergence. For :

Note

When a power series is differentiated: At the boundaries of the range of convergence which is a closed interval, the convergence might be lost.

When a power series is integral: At the boundaries of the range of convergence which is an open interval, the convergence might occur.