I am a math aficionado unable to find out where is the error in this proof showing that a odd perfect number does not exist. May I get some help?
Suppose $X$ is a perfect number and odd.
At most, $X$ can be divided by all odd numbers until $X/3$:
$1 + 3 + ... + X/3 = X$
We can divide it by $X$, so we get:
$(1 + 3 + ... + X/3)/X = 1$
Therefore, at most, we got the sum of the inverse of all odd numbers until $X/3$, at it must be equal to 1:
$1/3 + 1/5 + 1/7 + 1/9 +...+ 1/(X/3) = 1$
But the sum of the inverse of the infinite series of all odd numbers is at most $1-pi/4$
Therefore, if the largest possible sum of divisors is smaller than one, any other is going to be small too. So there is not odd perfect number.
