application of Cauchy criterion for convergence

Without using the methods of the entry determining series convergence, we show that the real-term series

n=01n!= 1+11!+12!+

is convergentMathworldPlanetmathPlanetmath by using Cauchy criterion for convergence, being in in equipped with the usual absolute valueMathworldPlanetmathPlanetmathPlanetmath |.| as

Let ε be an arbitrary positive number.  For any positive integer n, we have


whence we can as follows.

|1(n+1)!++1(n+p)!| =1(n+1)!++1(n+p)!

The last inequalityMathworldPlanetmath is true for all positive integers p, when  n> 1-lbε.  Thus the Cauchy criterion implies that the series converges.

Title application of Cauchy criterion for convergence
Canonical name ApplicationOfCauchyCriterionForConvergence
Date of creation 2013-03-22 19:03:22
Last modified on 2013-03-22 19:03:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Example
Classification msc 40A05
Related topic RealNumber
Related topic GeometricSeries
Related topic LogarithmusBinaris
Related topic NapiersConstant