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andy crisan
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 Posted: Mon Mar 08, 2010 5:17 pm Post subject: Sir convergent |
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Fie _{n\geq 1} "(x_n)_{n\geq 1}") un sir marginit de numere reale. Stiind ca =0 "\lim_{n\to\infty}(x_{n+1}-x_{n})=0") sa se arate ca sirul este convergent.
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mihai++
Bernoulli
Joined: 28 Nov 2007
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| Posted: Wed Mar 10, 2010 3:15 pm Post subject: |
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cineva vreo idee?
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mychrom
Euclid
Joined: 08 Oct 2007
Posts: 16
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| Posted: Wed Mar 10, 2010 3:31 pm Post subject: |
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Sper sa nu gresesc, dar daca luam  "x_n=\sin(H_n)") , unde  este seria armonica pana la n, se verifica conditiile din enunt, dar  nu are limita.
In primul rand, sirul este evident marginit, iar } cos (H_n+\frac{1}{2(n+1)}) "x_{n+1}-x_n=2\sin \frac{1}{2(n+1)} cos (H_n+\frac{1}{2(n+1)})") , deci =0 "\lim( x_{n+1}-x_n)=0") .
Pe de alta parte, pentru un  fixat, destul de mic, pot gasi o infinitate de perechi n si k, numere naturale, astfel incat  si, de asemenea, o infinitate de perechi m si t astfel incat }{2}\pi|<a "|\sin H_m-\frac{(2t+1)}{2}\pi|<a") , deci sirul nu are limita (de fapt, multimea valorilor aderente ale lui este [-1,1]). |
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turcas
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Joined: 28 Sep 2007
Posts: 83
Location: Simleu Silvaniei, jud Salaj
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| Posted: Wed Mar 10, 2010 3:38 pm Post subject: Re: Sir convergent |
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| andy crisan wrote: | Fie un sir marginit de numere reale. Stiind ca sa se arate ca sirul este convergent.
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Tudor are dreptate.
Ne folosim de faptul ca seria  este divergenta.
Deci pentru fiecare  natural, gasim un  astfel incat :
 . Asta ne asigura ca putem alege  si un subsir  "(x_{n_k})") al sirului  , astfel incat  si  . |
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Beniamin Bogosel
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Joined: 07 Mar 2008
Posts: 709
Location: Timisoara sau Sofronea (Arad)
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| Posted: Wed Mar 10, 2010 8:50 pm Post subject: |
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Incercati sa demonstrati ca daca =0 "\lim_{n\to \infty} (x_n-x_{n+1})=0") atunci multimea punctelor limita ale sirului  "(x_n)") este un interval compact.
Poate fi folositoare, ca o lema, desi nu cred ca ajuta in problema data.
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andy crisan
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Joined: 28 Dec 2008
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Location: Pitesti
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| Posted: Thu Mar 11, 2010 3:35 pm Post subject: |
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| Asa e problema e gresita. Cred ca mai bine o puneam la intrebari teoretice. |
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