Can anybody help me with the proof of this inequality please?
$\displaystyle sum_{k=1}^n(a_k).sum_{k=1}^n(1/a_k)<=n^2$
$\displaystyle a_k>0$
thank you
The correct inequality is this:
$\displaystyle \left(\sum_{k=1}^na_k\right)\left(\sum_{k=1}\frac{ 1}{a_k}\right)\geq n^2$
Use the AM-GM inequality.
$\displaystyle \sum_{k=1}^na_k\geq n\sqrt[n]{a_1a_2\ldots a_n}$
$\displaystyle \sum_{k=1}^n\frac{1}{a_k}\geq n\cdot\frac{1}{\sqrt[n]{a_1a_2\ldots a_n}}$
Now multiply the inequalities, member with member.