Question

Demonstrati ca 1 + 1 / radical 2 + 1/ radical 3 + 1/ radical 4 +...+ 1 / radical 2015 <90

Answer 1

[tex] \sqrt{n+1} - \sqrt{n} = \frac{( \sqrt{n+1} - \sqrt{n}) *( \sqrt{n+1} + \sqrt{n} ))}{ \sqrt{n+1} + \sqrt{n} } = \\ = \frac{1}{\sqrt{n+1} + \sqrt{n}} \ \textgreater \ \frac{1}{\sqrt{n+1} + \sqrt{n+1}} = \frac{1}{2 \sqrt{n+1} } =\ \textgreater \ \\ \frac{1}{ \sqrt{n+1} } \ \textless \ 2( \sqrt{n+1} - \sqrt{n}) \\ n=0=\ \textgreater \ \frac{1}{ \sqrt{1} } \ \textless \ 2( \sqrt{1} - \sqrt{0}) \\ n=1=\ \textgreater \ \frac{1}{ \sqrt{2} } \ \textless \ 2( \sqrt{2} - \sqrt{1}) \\ .... \\ n=2015=\ \textgreater \ \frac{1}{ \sqrt{2015} } \ \textless \ 2( \sqrt{2015} - \sqrt{2014}) \\ S\ \textless \ 2* \sqrt{2015} =\ \textgreater \ S\ \textless \ \sqrt{8060}\ \textless \ 90. [/tex]