Shortlist ONM 2010 pb 23

[Laurentiu Tucaa]

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]

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]

Putem oare integra prin parti?

[Marius Mainea]

Nu, evident

[Laurentiu Tucaa]

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 \( -f^{\prime}(1) \)

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