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harmonic number
The harmonic number of order of is defined as
Note that may be equal to , provided .
If , while , the harmonic series does not converge and hence the harmonic number does not exist.
If , we may just write as (this is a common notation).
Properties
-
If and then the sum is the Riemann zeta function.
-
If , then we get what is known simply as“the harmonic number”, and it has many important properties. For example, it has asymptotic expansion where is Euler’s constant.
Defines:
harmonic number of order
Related:
Series, AbsoluteConvergence, HarmonicSeries, PrimeHarmonicSeries, WolstenholmesTheorem
Type of Math Object:
Definition
Major Section:
Reference
Mathematics Subject Classification
26A06 One-variable calculus40A05 Convergence and divergence of series and sequences
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