(HMMT 2004) For every positive integer $n$, prove that

$

frac{sigma(1)}{1}+frac{sigma(2)}{2}+dots+frac{sigma(n)}{n} leq 2 n

$

If $d$ is a divisor of $i,$ then so is $frac{i}{d},$ and $frac{i / d}{i}=frac{1}{d} .$

Summing over all divisors $d$ of $i$ (which is $sigma(i)$ ), we see that $frac{sigma(i)}{i}$ is the sum of all the reciprocals of the divisors of $i ;$

that is,

$

frac{sigma(i)}{i}=sum_{d | i} frac{1}{d}

$ …..

**But how they concluded that $
frac{sigma(i)}{i}=sum_{d | i} frac{1}{d}
$** ??? ***

i am getting trouble in understanding this question in recent few days, what i till now understand is that

if

$d$ is divisor of $i$ then $i/d$ is also divisor of $i$ , so

$frac{i / d}{i}=frac{1}{d} .$ but they are not all divisors of i so how we get *** ,i also tried to take some examples but still can’t get it,

can anyone explain this ..

thankyou