# A New Approximation Formula for Computing the N-th Harmonic Number (Update)

 November 11, 2011 A New Approximation Formula for Computing the N-th Harmonic Number  (Update) A new approximate formula of giving more digits of accuracy for computing the n-th Harmonic Number is found as follows: ,  whereis Euler constant and n is a positive integer.

Table - Computation of Exact and Approximation Harmonic Numbers
 n (Exact) (Approximation) 1 1 1.0000364756158384 2 1.5 1.500001060257485 3 1.8333333333333333 1.8333334197475766 4 2.083333333333333 2.0833333459100944 5 2.283333333333333 2.2833333359731323 6 2.45 2.4500000007120852 7 2.5928571428571425 2.5928571430876115 8 2.7178571428571425 2.7178571429427802 9 2.8289682539682537 2.828968254003711 10 2.9289682539682538 2.928968253984269 11 3.0198773448773446 3.0198773448851135 12 3.103210678210678 3.1032106782146784 13 3.180133755133755 3.1801337551359232 14 3.251562326562327 3.2515623265635525 15 3.3182289932289937 3.3182289932297135 16 3.3807289932289937 3.3807289932294307 17 3.439552522640758 3.4395525226410317 18 3.4951080781963135 3.4951080781964894 19 3.547739657143682 3.547739657143797 20 3.597739657143682 3.5977396571437588 50 4.499205338329423 4.499205338329425 100 5.187377517639621 5.18737751763962 150 5.591180588643881 5.591180588643878 200 5.878030948121446 5.8780309481214434

In-Text or Website Citation
Tue N. Vu, A New Approximation Formula for Computing the N-th Harmonic Number (Update), 11/11/2011, from Series Math Study Resource.