Infinity Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! ====Real analysis==== In [[real analysis]], the symbol <math>\infty</math>, called "infinity", is used to denote an unbounded [[limit of a function|limit]].<ref>{{harvnb|Taylor|1955|loc=p. 63}}</ref> The notation <math>x \rightarrow \infty</math> means that ''<math>x</math>'' increases without bound, and <math>x \to -\infty</math> means that ''<math>x</math>'' decreases without bound. For example, if <math>f(t)\ge 0</math> for every ''<math>t</math>'', then<ref>These uses of infinity for integrals and series can be found in any standard calculus text, such as, {{harvnb|Swokowski|1983|pp=468β510}}</ref> * <math>\int_{a}^{b} f(t)\, dt = \infty</math> means that <math>f(t)</math> does not bound a finite area from <math>a</math> to <math>b.</math> * <math>\int_{-\infty}^{\infty} f(t)\, dt = \infty</math> means that the area under <math>f(t)</math> is infinite. * <math>\int_{-\infty}^{\infty} f(t)\, dt = a</math> means that the total area under <math>f(t)</math> is finite, and is equal to <math>a.</math> Infinity can also be used to describe [[infinite series]], as follows: * <math>\sum_{i=0}^{\infty} f(i) = a</math> means that the sum of the infinite series [[convergent series|converges]] to some real value <math>a. </math> * <math>\sum_{i=0}^{\infty} f(i) = \infty</math> means that the sum of the infinite series properly [[divergent series|diverges]] to infinity, in the sense that the partial sums increase without bound.<ref>{{Cite web|url=http://mathonline.wikidot.com/properly-divergent-sequences|title=Properly Divergent Sequences - Mathonline|website=mathonline.wikidot.com|access-date=2019-11-15}}</ref> In addition to defining a limit, infinity can be also used as a value in the extended real number system. Points labeled <math>+\infty</math> and <math>-\infty</math> can be added to the [[topological space]] of the real numbers, producing the two-point [[compactification (mathematics)|compactification]] of the real numbers. Adding algebraic properties to this gives us the [[extended real number]]s.<ref>{{citation | last1 = Aliprantis | first1 = Charalambos D. | last2 = Burkinshaw | first2 = Owen | edition = 3rd | isbn = 978-0-12-050257-8 | location = San Diego, CA | mr = 1669668 | page = 29 | publisher = Academic Press, Inc. | title = Principles of Real Analysis | url = https://books.google.com/books?id=m40ivUwAonUC&pg=PA29 | year = 1998 | url-status=live | archive-url = https://web.archive.org/web/20150515120230/https://books.google.com/books?id=m40ivUwAonUC&pg=PA29 | archive-date = 2015-05-15 }}</ref> We can also treat <math>+\infty</math> and <math>-\infty</math> as the same, leading to the [[one-point compactification]] of the real numbers, which is the [[real projective line]].<ref>{{harvnb|Gemignani|1990|loc=p. 177}}</ref> [[Projective geometry]] also refers to a [[line at infinity]] in plane geometry, a [[plane at infinity]] in three-dimensional space, and a [[hyperplane at infinity]] for general [[Dimension (mathematics and physics)|dimensions]], each consisting of [[Point at infinity|points at infinity]].<ref>{{citation|first1=Albrecht|last1=Beutelspacher|first2=Ute|last2=Rosenbaum|title=Projective Geometry / from foundations to applications|year=1998|publisher=Cambridge University Press|isbn=978-0-521-48364-3|page=27}}</ref> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page