Sequences of Functions

We know a sequence is a function that maps natural numbers to real or complex numbers. But we can also define a sequence of functions and therefore a sequence of sequences.

Some weird complicated stuff.

This leads to us being able to swap the limit and the sum signs under some condition?

This is then used to prove that the exponential series converges to \(e^z\) for all \(z \in \mathbb{C}\).

Convergence of Function Sequences

This is all a bit weird and she uses this stuff to define the trigonometric functions?

Pointwise Convergence

Uniform Convergence

She later on also shows a third type?

Normal Convergence

Is this the third type?