Series of functions

A series of functions on a set $X$ is an expression $\sum_{n=0}^\infty f_n$ where each $f_n:X\to\mathbb{R}$ (or $X\to\mathbb{C}$). Its partial sums are the functions

The series is said to converge (pointwise or uniformly) if the sequence $(s_N)$ converges in the corresponding sense to a limit function $s$.

This is the function-level analogue of a numerical series , with partial sums taken pointwise in $x$. Many criteria for uniform convergence of series are packaged as theorems about sequences $(s_N)$, such as the Weierstrass M-test .

Examples:

• On $(-1,1)$, the power series $\sum_{n=0}^\infty x^n$ has partial sums $s_N(x)=(1-x^{N+1})/(1-x)$ and converges pointwise to $1/(1-x)$.
• On $\mathbb{R}$, the series $\sum_{n=1}^\infty \frac{\sin(nx)}{n^2}$ converges uniformly by comparison with $\sum_{n=1}^\infty 1/n^2$ (an application of the M-test ).