PROBLEM NO.33(B0MC-2,DAY-5)

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For every positive integer $n$,prove that
\[\sum _{i=1}^{n}\frac{\sigma (i)}{i}\leqslant 2n \].

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Hint:
(i)

\[\sum _{d|n}\frac{1}{d}= \frac{\sigma (n)}{n}\]

(ii)

Try to show that \[\sum _{k=1}^{n}\frac{1}{k^{2}}< 2\]