Fie \( (x_{n})_{n\geq 1} \) un sir convergent de numere reale astfel incat sa existe
\( \lim_{n\to\infty}n\left(\frac{x_{n}}{x_{n-1}}-1\right)\in\mathbb{R}^{*} \).
Sa se arate ca \( \lim_{n\to\infty}x_{n}=0 \).
Sir convergent plus o conditie, atunci limita sa este 0
Moderators: Bogdan Posa, Laurian Filip, Beniamin Bogosel, Radu Titiu, Marius Dragoi
-
Cezar Lupu
- Site Admin
- Posts: 612
- Joined: Wed Sep 26, 2007 2:04 pm
- Location: Bucuresti sau Constanta
- Contact:
Sir convergent plus o conditie, atunci limita sa este 0
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
-
Bogdan Cebere
- Thales
- Posts: 145
- Joined: Sun Nov 04, 2007 1:04 pm
Re: sir convergent atunci limita sa este egala cu 0
Cum exista \( \lim_{n\to\infty}n\left(\frac{x_{n}}{x_{n-1}}-1\right)\in\mathbb{R}^{*} \), rezulta ca exista un M astfel incat \( n\left(\frac{x_{n}}{x_{n-1}}-1\right)<M \), adica \( \frac{x_{n}}{x_{n-1}}<\frac{M}{n}+1 \).Cezar Lupu wrote:Fie \( (x_{n})_{n\geq 1} \) un sir convergent de numere reale astfel incat sa existe
\( \lim_{n\to\infty}n\left(\frac{x_{n}}{x_{n-1}}-1\right)\in\mathbb{R}^{*} \).
Sa se arate ca \( \lim_{n\to\infty}x_{n}=0 \).
Trecand inegalitatea la limita avem ca \( \lim_{n\to\infty}\left(\frac{x_{n}}{x_{n-1}}\right)<1 \), deci \( \lim_{n\to\infty}{x_{n}}=0 \).