November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
Content preview from Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
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B.26 HARMONIC NUMBERS
Definition: Harmonic Numbers The generalized harmonic numbers {Hn(v)} are defined by the following finite sum:
(B.140) 
for 
and 
.
The Riemann zeta function is obtained when 
. The ordinary harmonic numbers are given by 
, of which the first 10 values are H1 = 1, H2 = 3/2, H3 = 11/6, H4 = 25/12, H5 = 137/60, H6 = 441/180, H7 = 3267/1260, H8 = 6849/2520, H9 = 64, 161/22, 680, and H10 = 66, 429/22, 680. The harmonic numbers are related to the Bernoulli numbers as follows:
(B.141) 
where γ is the Euler–Mascheroni constant described in Section B.27. From this expansion, we find that 
for large n.
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