## You are here

Homesum function of series

## Primary tabs

# sum function of series

Let the terms of a series be real functions $f_{n}$ defined in a certain subset $A_{0}$ of $\mathbb{R}$; we can speak of a function series. All points $x$ where the series

$\displaystyle f_{1}+f_{2}+\cdots$ | (1) |

converges form a subset $A$ of $A_{0}$, and we have the sum function $S\!:x\mapsto S(x)$ of (1) defined in $A$.

If the sequence $S_{1},\,S_{2},\,\ldots$ of the partial sums $S_{n}=f_{1}\!+\!f_{2}\!+\cdots+\!f_{n}$ of the series (1) converges uniformly in the interval $[a,\,b]\subseteq{A}$ to a function $S\!:x\mapsto S(x)$, we say that the series converges uniformly in this interval. We may also set the direct

Definition. The function series (1), which converges in every point of the interval $[a,\,b]$ having sum function $S:x\mapsto S(x)$, converges uniformly in the interval $[a,\,b]$, if for every positive number $\varepsilon$ there is an integer $n_{\varepsilon}$ such that each value of $x$ in the interval $[a,\,b]$ satisfies the inequality

$|S_{n}(x)-S(x)|<\varepsilon$ |

when $n\geqq n_{\varepsilon}$.

Note. One can without trouble be convinced that the term functions of a uniformly converging series converge uniformly to 0 (cf. the necessary condition of convergence).

The notion of uniform convergence of series can be extended to the series with complex function terms (the interval is replaced with some subset of $\mathbb{C}$). The significance of the uniform convergence is therein that the sum function of a series with this property and with continuous term-functions is continuous and may be integrated termwise.

## Mathematics Subject Classification

26A15*no label found*40A30

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections