Submitted by admin on Fri, 11/11/2011 - 7:20pm
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), from Series Math Study Resource.
- Printer-friendly version
- 234 reads