To find the sum of this series, we can use the formula for the harmonic series:

H_n = 1 + 1/2 + 1/3 + ... + 1/n = ln(n) + γ + O(1/n)

where H_n is the nth harmonic number, ln(n) is the natural logarithm of n, γ is the Euler-Mascheroni constant, and O(1/n) represents the error term.

In this case, we want to find the sum of the series from 1/1 to 1/100, which is equivalent to the 100th harmonic number minus 1:

A = H_100 - 1 = ln(100) + γ + O(1/100) - 1

Using a calculator, we can find the approximate value of ln(100) to be 4.60517 and the Euler-Mascheroni constant to be approximately 0.57721. Therefore, the sum of the series A is approximately:

A ≈ 4.60517 + 0.57721 - 1 ≈ 4.18238

So, the sum of the series A is approximately 4.18238.