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Forum Studenti •Dal rapporto -> radice a Cesàro
Pagina 1 di 1

Dal rapporto -> radice a Cesàro

Inviato: sabato 2 gennaio 2016, 10:01
da francicko
Sì potrebbe risolvere il seguente limite di successione [tex]((2n)!)/(n!)^2)^{1/n})[/tex], ricorrendo al confronto con l'integrale di una somma?
Ci ho provato mettendolo nella forma [tex]e^{(1/n)log(((2n)!)/(n!))}[/tex];
Ho provato ma ottengo un risultato errato [tex]e^{log2}=2[/tex], mentre il risultato corretto del limite e' [tex]4[/tex], boh magari non si può fare :roll:

[EDIT by Massimo Gobbino] Ho cambiato il titolo perché il precedente era troppo generico.

Re: limite

Inviato: sabato 2 gennaio 2016, 10:45
da GIMUSI
ma perché vuoi soffrire!?!?...questo è telefonatissimo per un criterio rapporto->radice (vd. allegato) :roll:

oppure in tal modo lo trovi noioso e vuoi sperimentare altre strade?..non dico che non sia interessante e forse anche istruttivo però magari specificalo se no finisco per postarti delle cose che trovi banali

ciao e buon anno :wink:

Re: limite

Inviato: sabato 2 gennaio 2016, 11:56
da francicko
x@GIMUSI.
Premetto che trovo sempre ed in ogni caso molto interessanti le tue risposte, ed apprezzo molto le spiegazioni precise ed istruttive che dai!!
In questo caso specifico volevo sperimentare la via del confronto con una somma integrale, solo che ottengo un risultato errato e non capisco il perche';
Puoi aiutarmi, magari descrivendo il procedimento usando il confronto con un integrale come su indicato?
Grazie!

Re: limite

Inviato: domenica 3 gennaio 2016, 16:11
da GIMUSI
ti ringrazio per la stima che ricambio e allego uno svolgimento del limite secondo tre metodi:
1 - criterio rapporto->radice
2 - approssimazione di stirling
3 - confronto serie-integrali

per il "3" ho tagliato un po' i passaggi ma si dovrebbe capire (nel caso basta vedere quelli degli esercizi nel recente thread "calcolo limiti" del 21/12/15)

Re: limite

Inviato: lunedì 4 gennaio 2016, 19:56
da Massimo Gobbino
@francicko, come dice GIMUSI quel limite è un telefonatissimo rapporto -> radice.

Poi ovviamente possiamo divertirci ad osservare che, fatti i logaritmi, il rapporto -> radice diventa un Cesàro, e quasi tutti i Cesàro si possono interpretare in termini di integrali e stime su primitive. Nel caso specifico

[tex]\displaystyle\log\sqrt[n]{\frac{(2n)!}{(n!)^2}}=\frac{1}{n}\sum_{k=1}^{n}\log\frac{2k(2k-1)}{k^2}[/tex]

L'ultima sommatoria la possiamo riscrivere come

[tex]\displaystyle\frac{1}{n}\sum_{k=1}^{n}\left[\log 4+\log\left(1-\frac{1}{2k}\right)\right][/tex]

e da qui volendo si chiude in vario modo, compreso stimare le sommatorie con integrali.