example of converging increasing sequence

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

 $x_{1}:=\sqrt[q]{a},$
 $x_{2}:=\sqrt[q]{a+x_{1}}=\sqrt[q]{a+\sqrt[q]{a}},$
 $x_{3}:=\sqrt[q]{a+x_{2}}=\sqrt[q]{a+\sqrt[q]{a+\sqrt[q]{a}}},$

and generally

 $\displaystyle x_{n}:=\sqrt[q]{a+x_{n-1}}.$ (1)

Since  $x_{1}>0$,  the two first above equations imply that  $x_{1}.  By induction on $n$ one can show that

 $x_{1}

The numbers $x_{n}$ are all below a finite bound $M$.  For demonstrating this, we write the inequality$x_{n}  in the form  $x_{n}<\sqrt[q]{a+x_{n}}$, which implies   $x_{n}^{q},  i.e.

 $\displaystyle x_{n}^{q}-x_{n}-a<0$ (2)

for all $n$.  We study the polynomial

 $f(x):=x^{q}-x-a=x(x^{q-1}-1)-1.$

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

 $\displaystyle x^{q}-x-a=0$ (3)

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

The proven facts

 $x_{1}

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

 $x_{1},\,x_{2},\,x_{3},\,\ldots,\,x_{n},\,\ldots$

converges to a limit $x^{\prime}\leqq M$.

Taking limits of both sides of (1) we see that $x^{\prime}=\sqrt[q]{a+x^{\prime}}$,  i.e.  $x^{\prime q}-x^{\prime}-a=0$,  which means that  $x^{\prime}=M$,  in other words: the limit of the sequence is the only $M$ of the equation (3).

References

• 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos.  Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
Title example of converging increasing sequence ExampleOfConvergingIncreasingSequence 2013-03-22 17:40:44 2013-03-22 17:40:44 pahio (2872) pahio (2872) 6 pahio (2872) Example msc 40-00 NthRoot BolzanosTheorem