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# 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 $x_{1}<x_{2}$. By induction on $n$ one can show that

$x_{1}<x_{2}<x_{3}<\ldots<x_{n}<\ldots$ |

The numbers $x_{n}$ are all below a finite bound $M$. For demonstrating this, we write the inequality $x_{n}<x_{{n+1}}$ in the form $x_{n}<\sqrt[q]{a+x_{n}}$, which implies $x_{n}^{q}<a+x_{n}$, 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 positive root $x=M>1$, and that $f$ is negative for $0<x<1$ and positive for $x>M$. Thus we can conclude by (2) that $x_{n}<M$ for all values of $n$.

The proven facts

$x_{1}<x_{2}<x_{3}<\ldots<x_{n}<\ldots<M$ |

settle, by the theorem of the parent entry, that the sequence

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

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 positive root $M$ of the equation (3).

# References

- 1 E. Lindelöf: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).

## Mathematics Subject Classification

40-00*no label found*

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## Comments

## Another example

Jussi,

You might take a look at "proof that Euler's constant exists" for another example of such a series used in "real life".

Roger

## Re: Another example

Thanks, Roger,

I added a note pointing to your proof application.

Jussi