Matematické Fórum

vanok · 06. 05. 2013 19:07 — Editoval vanok (07. 05. 2013 00:41)

Hello
Here is a beautiful exercise:

Let be $S$ the set of all the increasing sequences of integer $(u_n)_{n \in \mathbb{N}}$ of first term $u_0$ upper or equal to 2.
We note $v_n= \frac 1{u_o u_1...u_n}$ and $x_n= \sum_{i=0}^{n} v_i$
a)Show that the sequence $(x_n)$ is increasing and majored.
Deduct from it that $(x_n)$ converges on the real x such as $0< x \leq 1$ .
b)Show that the application $f:S \rightarrow ]0;1]:(u_n) \mapsto x$ is a bijection.
c) Show that x is rational iff the sequence $(u_n)$ of which hi is the image is bounded

Pavel · 06. 05. 2013 22:20 — Editoval Pavel (06. 05. 2013 22:22)

↑ vanok:

Autor skryl část textu. Zobrazit/skrýt!

vanok · 07. 05. 2013 00:50 — Editoval vanok (07. 05. 2013 01:22)

Hi ↑ Pavel:
observation: an increasing sequence is not necessarily strictly increasing sequence

HINT:
Very quickly
a)
$v_n\leq \frac 1{2^{n+1}}$
so
$x_n < 1$
b) reciprocally
We can use
$a_0=x$ and $u_0= [\frac 1 {a_0}] +1$
$a_{k+1} =u_k a_k-1$ and $u_{k+1}= [\frac 1 {a_{k+1}}] +1$ ....

Matematické Fórum

#1 06. 05. 2013 19:07 — Editoval vanok (07. 05. 2013 00:41)

aproximation

#2 06. 05. 2013 22:20 — Editoval Pavel (06. 05. 2013 22:22)

Re: aproximation

#3 07. 05. 2013 00:50 — Editoval vanok (07. 05. 2013 01:22)

Re: aproximation

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