Math Help - Using integrals :D

1. Using integrals :D

${{\lim_{\substack{n\rightarrow\infty}} {\left( \frac {1}{\sqrt{n}\sqrt{n+1}}+\frac {1}{\sqrt{n}\sqrt{n+2}}+...+\frac {1}{\sqrt{n}\sqrt{n+n}}\right)}$

2. Originally Posted by rondo09
${{\lim_{\substack{n\rightarrow\infty}} {\left( \frac {1}{\sqrt{n}\sqrt{n+1}}+\frac {1}{\sqrt{n}\sqrt{n+2}}+...+\frac {1}{\sqrt{n}\sqrt{n+n}}\right)}$
${\lim_{n\rightarrow\infty}( \frac {1}{\sqrt{n}\sqrt{n+1}}+\frac {1}{\sqrt{n}\sqrt{n+2}}+...+\frac {1}{\sqrt{n}\sqrt{n+n}})$

= ${\lim_{n\rightarrow\infty}\sum_{i=1}^{n}( \frac {1}{\sqrt{n}\sqrt{n+i}})$

= ${\lim_{n\rightarrow\infty}\sum_{i=1}^{n}(\frac{1}{ \sqrt{1+\frac{i}{n}}})(\frac{1}{n})$

= $\int_{0}^{1}\frac{dx}{\sqrt{1+x}}$

= $2\sqrt{1+x}\big|_0^{1}$

= $2(\sqrt{2}-1)$