Actually, the sum of all positive even numbers is -1/6.
Now, traditionally, the sum from 1 to infinity of 2n does not converge, and thus should not even exist. However, there are ways to get around this inconvenient fact. These methods use analytic functions of complex variables to calculate values of divergent series by analogy with convergent series.
Now, to calculate the sum of all positive even numbers, it will be easier to calculate the sum of all positive integers, which is just one-half of the previous sum. But this, too, is divergent. Let us generalize the equation in such a way that variants of the equation will converge, even if the original does not.
The value we will use is the sum from n equals one to infinity of n to the -s power, where s is an arbitrary complex number. This is just the Riemann zeta function, a well-studied function from number theory. The sum of all positive integers is just ζ(-1), and from this, we can determine the sum of all positive even numbers to be 2ζ(-1). Luckily for us, although the traditional definition of the Riemann zeta function exists only for Re(s)>1, the analytic extension of this function exists for all s≠1, meaning that all such sequences other than the harmonic sequence have some equivalent of the limiting value.
The calculation of ζ(-1) is fairly complex, but luckily for us, it has already been done, and we know that it is -1/12. Multiplying by two, the sum of all positive even numbers is -1/6, and not -2 as this writeup erroneously claims.