prime theorem of a convergent sequence, a


Suppose (an) is a positive real sequenceMathworldPlanetmath that converges to L. Then the sequence of arithmetic meansMathworldPlanetmath (bn)=(n-1k=1nak) and the sequence of geometric meansMathworldPlanetmath (cn)=(a1ann) also converge to L.


We first show that (bn) converges to L. Let ε>0. Select a positive integer N0 such that nN0 implies |an-L|<ε/2. Since (an) converges to a finite value, there is a finite M such that |an-L|<M for all n. Thus we can select a positive integer NN0 for which (N0-1)M/N<ε/2.

By the triangle inequalityMathworldMathworldPlanetmath,

|bn-L| 1nk=1n|ak-L|

Hence (bn) converges to L.

To show that (cn) converges to L, we first define the sequence (dn) by dn=cnn=a1an. Since dn is a positive real sequence, we have that

lim infdn+1dnlim infdnnlim supdnnlim supdn+1dn,

a proof of which can be found in [1]. But dn+1/dn=an+1, which by assumptionPlanetmathPlanetmath converges to L. Hence dnn=cn must also converge to L. ∎


Title prime theorem of a convergent sequence, a
Canonical name PrimeTheoremOfAConvergentSequenceA
Date of creation 2013-03-22 14:49:45
Last modified on 2013-03-22 14:49:45
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 24
Author georgiosl (7242)
Entry type Theorem
Classification msc 40-00
Related topic ArithmeticMean
Related topic GeometricMean