Shortlist ONM 2010 pb 23

Laurentiu Tucaa · Post by **Laurentiu Tucaa** » Tue Apr 13, 2010 7:50 pm

Fie \( f:[0,1]\rightarrow\mathbb{R} \) integrabila,derivabila in 1 si \( f(1)=0 \).Aratati ca \( \lim_{n\to\infty} n^2\int_0^1 x^nf(x)dx=-f^{\prime}(1) \)

Dan Stefan Marinescu,Viorel Cornea

Marius Mainea · Post by **Marius Mainea** » Tue Apr 13, 2010 8:17 pm

Integram prin parti, apoi folosim Propozitia

Daca \( g:[0,1]\longrightarrow\mathbb{R} \) este integrabila si continua in \( x=1 \), atunci \( \lim_{n\to\infty}n\int_0^1x^ng(x)dx=g(1) \)

Theodor Munteanu · Post by **Theodor Munteanu** » Wed Apr 14, 2010 4:11 pm

Putem oare integra prin parti?

Marius Mainea · Post by **Marius Mainea** » Thu Apr 15, 2010 9:12 pm

Nu, evident

Laurentiu Tucaa · Post by **Laurentiu Tucaa** » Mon Sep 13, 2010 2:38 pm

Problemuta e destul de simpla .Primul pas se ia functia \( g:[0,1]\rightarrow\mathbb{R},g(x)=\frac{f(x)-f(1)}{x-1} \) ,care este evident marginita .Mai mult \( n^2\int_0^{1-\frac{1}{\sqrt{n}}}x^ng(x)(x-1)dx\rightarrow 0 \) si ce ramane tinde spre \( -f^{\prime}(1) \)