convergence in the mean


Let

bn:=a1+a2++ann(n=1,2,3,)

be the arithmetic meanMathworldPlanetmath of the numbers a1,a2,,an.   The sequenceMathworldPlanetmath

a1,a2,a3, (1)

is said to converge in the mean (http://planetmath.org/ConvergenceInTheMean) iff the sequence

b1,b2,b3, (2)

converges.
On has the

Theorem.  If the sequence (1) is convergentMathworldPlanetmathPlanetmath having the limit A, then also the sequence (2) converges to the limit A.  Thus, a convergent sequence is always convergent in the mean.

Proof.  Let ε be an arbitrary positive number.  We may write

|A-bn| =|A-1n(a1++ak)-1n(ak+1++an)|
=|1n[(A-a1)++(A-ak)]+1n[(A-ak+1)++(A-an)]|
|(A-a1)++(A-ak)|n+|A-ak+1|++|A-an|n.

The supposition implies that there is a positive integer k such that

|A-ai|<ε2 for all i>k.

Let’s fix the integer k.  Choose the number l so great that

|(A-a1)++(A-ak)|n<ε2 for n>l.

Let now  n>max{k,l}.  The three above inequalitiesMathworldPlanetmath yield

|A-bn|<ε2+1n(n-k)ε2<ε2+ε2=ε,

whence we have

limnbn=A.

Note.  The converse (http://planetmath.org/Converse) of the theorem is not true.  For example, if

an:=1+(-1)n2

i.e. if the sequence (1) has the form  0,1,0,1,0,1,,  then it is divergent but converges in the mean to the limit 12; the corresponding sequence (2) is 0,12,13,24,25,36,37,48,49,

Title convergence in the mean
Canonical name ConvergenceInTheMean
Date of creation 2015-04-08 7:29:35
Last modified on 2015-04-08 7:29:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Definition