# 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

${x}_{n}:=\sqrt[q]{a+{x}_{n-1}}.$ | (1) |

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

$$ |

The numbers ${x}_{n}$ are all below a finite bound $M$. For demonstrating this, we write the inequality^{} $$ in the form $$, which implies $$, i.e.

$$ | (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 $\mathrm{\infty}$. Because $f$ as a polynomial function is also continuous^{}, we infer that the equation

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

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

The proven facts

$$ |

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

$${x}_{1},{x}_{2},{x}_{3},\mathrm{\dots},{x}_{n},\mathrm{\dots}$$ |

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 |
---|---|

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 |