To solve this problem, we need to find a pattern in the denominators of the fractions being added. Notice that all the denominators are multiples of 2, 3, or both. We can write each fraction with a common denominator of 24:

S = (12/24) + (8/24) + (6/24) + (4/24) + (3/24) + (2/24) + (1/24) + ...

Now we can see that each term in the sum is of the form k/24, where k is a positive integer. So we can rewrite the sum as:

S = (1/24) + (2/24) + (3/24) + (4/24) + (6/24) + (8/24) + (12/24) + ...

Notice that the terms in the sum are now in increasing order. We can use the formula for the sum of an arithmetic series to find the sum:

S = n/2 * (a + l)

where n is the number of terms, a is the first term, and l is the last term. In this case, we have:

n = 9
a = 1/24
l = 12/24 = 1/2

Substituting these values into the formula, we get:

S = 9/2 * (1/24 + 1/2)

Simplifying, we get:

S = 9/2 * 13/48

S = 39/32

Therefore, the sum of the series is 39/32.