Editing Infinitesimal (section)

=== Formal series ===

==== Laurent series ====
An example from category 1 above is the field of [[Laurent series]] with a finite number of negative-power terms. For example, the Laurent series consisting only of the constant term 1 is identified with the real number&nbsp;1, and the series with only the linear term&nbsp;''x'' is thought of as the simplest infinitesimal, from which the other infinitesimals are constructed. Dictionary ordering is used, which is equivalent to considering higher powers of&nbsp;''x'' as negligible compared to lower powers. [[David O. Tall]]<ref>{{cite web |url=http://www.jonhoyle.com/MAAseaway/Infinitesimals.html |title=Infinitesimals in Modern Mathematics |publisher=Jonhoyle.com |access-date=2011-03-11 |url-status=dead |archive-url=https://web.archive.org/web/20110713115815/http://www.jonhoyle.com/MAAseaway/Infinitesimals.html |archive-date=2011-07-13 }}</ref> refers to this system as the super-reals, not to be confused with the [[superreal number]] system of Dales and Woodin. Since a [[Taylor series]] evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus on transcendental functions if they are analytic. These infinitesimals have different first-order properties than the reals because, for example, the basic infinitesimal&nbsp;''x'' does not have a square root.

==== The Levi-Civita field ====
The [[Levi-Civita field]] is similar to the Laurent series, but is algebraically closed. For example, the basic infinitesimal x has a square root. This field is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented in floating-point.<ref>{{Cite web|url=http://www.uwec.edu/surepam/media/RS-Overview.pdf|title=Analysis on the Levi-Civita Field, a Brief Overview|last=Shamseddine|first=Khodr|archive-url=https://web.archive.org/web/20110608043202/http://www.uwec.edu/surepam/media/RS-Overview.pdf|archive-date=2011-06-08|url-status=dead}}</ref>

==== Transseries ====
The field of [[transseries]] is larger than the Levi-Civita field.<ref>{{Cite journal|last=Edgar|first=Gerald A.|date=2010|title=Transseries for Beginners|url=https://people.math.osu.edu/edgar.2/preprints/trans_begin/|journal=[[Real Analysis Exchange]]|volume=35|issue=2|pages=253–310|doi=10.14321/realanalexch.35.2.0253|arxiv=0801.4877|s2cid=14290638}}</ref> An example of a transseries is:

:<math>e^\sqrt{\ln\ln x}+\ln\ln x+\sum_{j=0}^\infty e^x x^{-j},</math>

where for purposes of ordering ''x'' is considered infinite.