To solve this problem, we need to find a common denominator for the fractions on both sides of the equation.

The sum of the fractions on the left side can be expressed as:

1/2 + 1/3 + 1/4 + ... + 1/19 + 1/20 = (1/2 + 1/3 + 1/4 + ... + 1/19) + 1/20

This is a harmonic series, which can be simplified as:

(1/2 + 1/3 + 1/4 + ... + 1/19) = H(20) - 1

Where H(20) is the 20th harmonic number. The formula for the nth harmonic number is:

H(n) = 1 + 1/2 + 1/3 + ... + 1/n

So, H(20) = 1 + 1/2 + 1/3 + ... + 1/20

Now, let's simplify the sum of the fractions on the right side of the equation:

1/19 + 2/18 + 3/17 + ... + 18/2 + 19/1 = (1 + 2 + 3 + ... + 18 + 19) / (19 * 18 * 17 * ... * 2 * 1)

The sum of the numbers from 1 to 19 is given by the formula for the sum of an arithmetic series:

S = n/2 * (first term + last term)

S = 19/2 * (1 + 19) = 190

Therefore, the sum of the fractions on the right side is:

190 / (19!)

Now, we can set up the equation:

(H(20) - 1) + 1/20 = 190 / (19!)

(H(20) - 1) + 1/20 = 190 / 121645100408832000

Now, we can solve for x by dividing both sides by the sum of the fractions on the left side:

x = (190 / 121645100408832000) / (H(20) - 1 + 1/20)

x ≈ 1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000