Întrebare Limita

Buna ziua
Sa se calculeze :
\[\lim_{n\rightarrow \infty}\dfrac{1}{n\sqrt{n}}\cdot \sum_{k=1}^n\dfrac{k}{\sqrt{n+k}}\]

Rezultatul poate fi:

\[a)2,b)2-\sqrt{2},c)\dfrac{4-2\sqrt{2}}{3},d)2+\sqrt{2},e)\dfrac{2+\sqrt{2}}{3}\]
multumesc

Buna seara,

\(\lim_{n\rightarrow\infty}\frac{1}{n\sqrt{n}}\sum_{k=1}^{n}\frac{k}{\sqrt{n+k}}=\lim_{n\rightarrow\infty}\frac{1}{n\sqrt{n}}\sum_{k=1}^{n}\frac{k}{\sqrt{n}\sqrt{1+\frac{k}{n}}}=\)
\(=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\frac{\frac{k}{n}}{\sqrt{1+\frac{k}{n}}}=\int_{0}^{1}\frac{x}{\sqrt{1+x}}dx\) , rezultat final c).

Buna seara
Am inteler multumesc!