Built-in Mathematica Symbol

HarmonicNumber

HarmonicNumber[n]
gives the n

harmonic number

.

HarmonicNumber[n, r]
gives the harmonic number

of order r.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

The harmonic numbers are given by with .

HarmonicNumber can be evaluated to arbitrary numerical precision.

HarmonicNumber automatically threads over lists.

EXAMPLES

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Basic Examples (3)

First ten harmonic numbers:

Plot harmonic numbers:

Carry out sums involving harmonic numbers:

First ten harmonic numbers:

In[1]:=

Out[1]=

Plot harmonic numbers:

In[1]:=

Out[1]=

Carry out sums involving harmonic numbers:

In[1]:=

Out[1]=

Scope (8)

Evaluate exact values at large arguments:

Non-integer arguments:

Complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

HarmonicNumber threads element-wise over lists and arrays:

Series expansion at any point:

TraditionalForm formatting:

Generalizations & Extensions (5)

Series expansion at infinity:

HarmonicNumber can be applied to power series:

Evaluate at exact arguments:

Series expansion at any point:

Series expansion at infinity:

Applications (4)

The average number of comparisons in Quicksort:

Plot over the complex plane:

Book stacking with the maximal overhang:

Picking the best candidate out of n after x evaluated choices [more info]:

Evaluate for n=100:

Plot as a function of harem size:

Properties & Relations (5)

Use FullSimplify to simplify expressions containing harmonic numbers:

Expand in simpler functions:

Sums:

Generate from sums and integrals:

Generating function:

Possible Issues (3)

Large arguments can give results too large to be computed explicitly:

Machine-number inputs can give high-precision results:

Often results are expressed in PolyGamma instead of HarmonicNumber: