Date: Nov 25, 2012 4:37 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Matheology § 162
Matheology § 162

About limits of real sequences.

The limit of an infinite sequence (a_k) of real numbers a_k is

determined solely by the finite terms of the sequence. Otherwise, the

limit would not have to be *computed* but would have to be *created*.

Analysis is concerned with analyzing, i.e., with finding.

To give an example, we can state with absolute certainty that in the

real numbers the sequence

0.1, 0.11, 0.111, ...

has the limit 0.111... = 1/9.

That is independent of the method which is used to analyze the

sequence. But there are different aspects of the limit, namely the

numerical value of the limit, the set of coefficients of the power

series, its cardinal number, the set of indexes which belong to a

digit 1, its cardinal number, the set of indexes which belong to a

digit 2, its cardinal number, the set of different digits appearing in

the limit, and many further aspects.

If any of these aspects is computed by another than the analytical

method and turns out as deviating from the analytical result, then the

other method is not suitable for analytical purposes.

Regards, WM