example of converging increasing sequence

Let a be a positive real number and q an integer greater than 1.  Set


and generally

xn:=a+xn-1q. (1)

Since  x1>0,  the two first above equations imply that  x1<x2.  By induction on n one can show that


The numbers xn are all below a finite bound M.  For demonstrating this, we write the inequalityMathworldPlanetmathxn<xn+1  in the form  xn<a+xnq, which implies   xnq<a+xn,  i.e.

xnq-xn-a<0 (2)

for all n.  We study the polynomialPlanetmathPlanetmath


From its latter form we see that the function f attains negative values when  0x1  and that f increases monotonically and boundlessly when x increases from 1 to .  Because f as a polynomial function is also continuousMathworldPlanetmath, we infer that the equation

xq-x-a=0 (3)

has exactly one root (http://planetmath.org/Equation)   x=M>1,  and that f is negative for  0<x<1  and positive for  x>M.  Thus we can conclude by (2) that  xn<M  for all values of n.

The proven facts


settle, by the theorem of the parent entry (http://planetmath.org/NondecreasingSequenceWithUpperBound), that the sequenceMathworldPlanetmath


converges to a limit xM.

Taking limits of both sides of (1) we see that x=a+xq,  i.e.  xq-x-a=0,  which means that  x=M,  in other words: the limit of the sequence is the only M of the equation (3).


  • 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos.  Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
Title example of converging increasing sequence
Canonical name ExampleOfConvergingIncreasingSequence
Date of creation 2013-03-22 17:40:44
Last modified on 2013-03-22 17:40:44
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Example
Classification msc 40-00
Related topic NthRoot
Related topic BolzanosTheorem