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A New Approximate Formula for Computing the N-th Harmonic Number

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.

 

 

Below is the computation table of some approximate and exact Harmonic Numbers.


Table - Computation of Exact and Approximate Harmonic Numbers

n  (Exact Computation)  (Approximate Computation)
1 1.0000364756158384
2 1.5                                (= 3/2) 1.500001060257485
3 1.8333333333333333  (= 11/6) 1.8333334197475766
4 2.083333333333333    (= 25/12) 2.0833333459100944
5 2.283333333333333    (= 137/60) 2.2833333359731323
6 2.45                              (= 49/20) 2.4500000007120852
7 2.5928571428571425  (= 363/140) 2.5928571430876115
8 2.7178571428571425  (= 761/280) 2.7178571429427802
9 2.8289682539682537  (= 7129/2520) 2.828968254003711
10 2.9289682539682538  (= 7381/2520) 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

(November 11, 2011)

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.