integral test


Consider a sequence (an)={a0,a1,a2,a3,} and given M consider any monotonically nonincreasing function f:[M,+) which extends the sequence, i.e.

f(n)=an  nM

An example is

an=2n    f(x)=2x

(the former being the sequence {0,2,4,6,8,} and the later the doubling function for any real number.

We are interested on finding out when the summation

n=0an

converges.

The integral testMathworldPlanetmath states the following.

The series

n=0an

converges if and only if the integral

Mf(x)𝑑x

is finite.

Title integral test
Canonical name IntegralTest
Date of creation 2013-03-22 12:27:12
Last modified on 2013-03-22 12:27:12
Owner drini (3)
Last modified by drini (3)
Numerical id 20
Author drini (3)
Entry type Theorem
Classification msc 40A05
Related topic Function
Related topic Sequence
Related topic Limit