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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
example of converging increasing sequence
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(Example)
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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
![$\displaystyle x_n := \sqrt[q]{a+x_{n-1}}.$](http://images.planetmath.org:8080/cache/objects/10118/js/img1.png "$\displaystyle x_n := \sqrt[q]{a+x_{n-1}}.$") |
(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.
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(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
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(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 $$x_1,\,x_2,\,x_3,\,\ldots,\,x_n,\,\ldots$$ converges to a limit $x' \leqq M$ .
Taking limits of both sides of (1) we see that $x' = \sqrt[q]{a+x'}$ , i.e. $x'^q-x'-a = 0$ , which means that $x' = M$ , in other words: the limit of the sequence is the only positive root $M$ of the equation (3).
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- E. LINDELÖF: Johdatus korkeampaan analyysiin. Neljäs painos. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
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"example of converging increasing sequence" is owned by pahio.
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Cross-references: sides, limit, converges, sequence, theorem, continuous, polynomial function, monotonically, negative, function, polynomial, inequality, bound, finite, numbers, induction, imply, equations, integer, real number, positive
This is version 3 of example of converging increasing sequence, born on 2007-12-10, modified 2007-12-11.
Object id is 10118, canonical name is ExampleOfConvergingIncreasingSequence.
Accessed 721 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
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Pending Errata and Addenda
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