---
tag: 0Y8FRYX8Q2
kind: concept
name: Sequence of functions
msc: [40A30]
uses: [0QE8STGJH2, 056FSKG32J, 08FPABDQ4C]
sources:
  - url: https://en.wikipedia.org/wiki/Function_series
    locator: Convergence
---

## Summary

A sequence of functions is a sequence $(f_n)$ of functions $f_n \colon S \to \mathbb{R}$ on a common set $S$. For each fixed $x \in S$, the values $f_n(x)$ form a [[sequence of real numbers|0QE8STGJH2]]. Such sequences can converge in several ways, mainly [[pointwise|056FSKG32J]] or [[uniformly|08FPABDQ4C]].

## Remarks

The two modes give different properties of the limit. A uniform limit of continuous functions is continuous, but a pointwise limit need not be.
Examples are the Taylor polynomials of a function, and the partial sums of a power series.

## Explanations

### L1 human

A sequence of functions is a list of graphs, one for each $n$.
We ask whether the graphs approach the graph of a limit function, and in what sense.