Sequences and Series

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 October 2009

Sequences

Let $X$ be a set. A sequence $(x_1,x_2,x_3,\dots )$ of points in $X$ is a function

$\begin{array}{ccc}{\mathbb{Z}}_{>0}& ⟶& X\\ n& ⟼& x_n\end{array}$

Let $X$ be a set and let $(x_1,x_2,x_3,\dots )$ be a sequence in $X$. A limit of the sequence $(x_1,x_2,x_3,\dots )$ is a limit point of the sequence with respect to the Fréchet filter on ${\mathbb{Z}}_{>0}$. Write

$y=\underset{n\to \infty }{\lim }{x}_n$ if $y$ is a limit of the sequence $(x_1,x_2,x_3,\dots )$

The sequence $(x_1,x_2,x_3,\dots )$ converges if $\underset{n\to \infty }{\lim }{x}_n$ exists and is unique.

The sequence $(x_1,x_2,x_3,\dots )$ diverges if it does not converge.

Let $X$ be a totally ordered set and let $(x_1,x_2,x_3,\dots )$ be a sequence in $X$. The upper limit of $(x_1,x_2,x_3,\dots )$ is

$\lim \sup x_n=\underset{n\to \infty }{\lim }\sup \{x_n,x_{n+1},x_{n+2},\dots \}$

Let $X$ be a totally ordered set and let $(x_1,x_2,x_3,\dots )$ be a sequence in $X$. The lower limit of $(x_1,x_2,x_3,\dots )$ is

$\lim \inf x_n=\underset{n\to \infty }{\lim }\inf \{x_n,x_{n+1},x_{n+2},\dots \}$

A sequence $(x_1,x_2,x_3,\dots )$ is bounded if the set $\{x_1,x_2,x_3,\dots \}$ has an upper bound.

A sequence $(x_1,x_2,x_3,\dots )$ is monotonically increasing if it satisfies if $i\in {\mathbb{Z}}_{>0}$ then $x_i\ge x_{i+1}$.

A sequence $(x_1,x_2,x_3,\dots )$ is monotonically decreasing if it satisfies if $i\in {\mathbb{Z}}_{>0}$ then $x_i\le x_{i+1}$.

Series

Let $X$ be a set and let $(x_1,x_2,x_3,\dots )$ be a sequence in $X$. The series $\sum _{n=1}^{\infty }{a}_n$ is

the sequence $(s_1,s_2,s_3,\dots )\text{where}{s}_k=a_1+a_2+\dots a_k.$

Write

$\sum _{n=1}^{\infty }{a}_n=a\text{if}\underset{n\to \infty }{\lim }{s}_n=a.$

The series $\sum _{n=1}^{\infty }{a}_n$ converges if the sequence $(s_1,s_2,s_3,\dots )$ converges.

The series $\sum _{n=1}^{\infty }{a}_n$ diverges if the sequence $(s_1,s_2,s_3,\dots )$ diverges.

The series $\sum _{n=1}^{\infty }{a}_n$ converges absolutely if the series $\sum _{n=1}^{\infty }\left|a_n\right|$ converges.

References

[BG] A. Braverman and D. Gaitsgory , Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)