In this article, we will look at what is meant by Series of Functions. For these series of functions, we will cover concepts such as pointwise and uniform convergence. Hence, we will also need to look at the so-called Weierstrass’ M-test. It is advised to read my article about infinite series and my article about pointwise and uniform convergence first.
Defining Series of Functions
Let us first formally define a series of functions. We have already talked about series before where we used real numbers. Now, we will do the same but with functions instead. Hence, we have:
Where we remember that:
Here, we can call these sub-parts partial sums. If we then consider these functions in a specific point, i.e., x = a, then it will give us an ordinary infinite series.
Convergence of a Series of Functions
Let us then consider how we can determine whether a series of functions converges, either pointwise or uniform. We can define it as the following:
Hence, we can see that the convergence of a series is defined by the convergence of its sequence of partial sums. Let us try to take an illustrative example. Imagine that we have the series:
