““Luckily, there's a neat approximation formula for just this sum. The higher the number n, the better the estimate will be. It is mathematically proven that as n grows to infinity, the approximation formula converges to the true value. Here it is: H(n) ≈ ln(n) + 0.58 The value 0.58 comes from rounding off the Euler-Mascheroni constant, which should be where the 0.58 is now. But since we just want to approximate, there's no need to be overly precise. For our purposes the rounded off value will do just fine.””
