Series of functions

Let $X$ be a set and let $(Y,d_Y)$ be a metric space (typically $Y=\mathbb{R}$ or $\mathbb{C}$). A series of functions on $X$ is a formal expression

where each $f_n:X\to Y$. The associated partial sums are the functions

One says the series converges pointwise (respectively, converges uniformly ) if $S_N$ converges pointwise (respectively, uniformly) to some function $S:X\to Y$.

Series of functions are used to define analytic expansions (power series, Fourier series) and to build functions by approximations. The mode of convergence determines which operations can be interchanged with summation.

Examples:

• On $(-1,1)$, $\sum_{n=0}^\infty x^n$ is a series of functions with partial sums $S_N(x)=\sum_{n=0}^N x^n$.
• On $\mathbb{R}$, $\sum_{n=1}^\infty \frac{\sin(nx)}{n^2}$ is a series of functions (Fourier-type).
• The Weierstrass M-test gives a standard sufficient condition for uniform convergence of such series.