Ulikhet

[Gustav]

Vis at $1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots + \frac{1}{n^2}<\frac74$ for alle naturlige tall n.

[Brahmagupta]

[tex]1+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}\cdots + \frac1{n^2} < 1+\frac14+\frac1{2\cdot3}+\frac1{3\cdot4}+\frac1{4\cdot5}+\cdots+\frac1{(n-1)n}[/tex]

[tex]=\frac54 +\frac12-\frac13+\frac13-\frac14+\cdots - \frac1{n-1}+\frac1{n-1}-\frac1{n}=\frac74-\frac1{n}<\frac74[/tex]