The series
 |
(1)
|
is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function
. The divergence, however, is very
slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme
(ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire
2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann
Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10).
Progressions of the form
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(2)
|
are also sometimes called harmonic series (Beyer 1987).
Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2,
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(3)
|
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(4)
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and since an infinite sum of 1/2's diverges, so does the harmonic series.
The generalization of the harmonic series
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(5)
|
is known as the Riemann zeta function.
The sum of the first few terms of the harmonic series is given analytically by the 
th harmonic
number
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(6)
|
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(7)
|
where 
is the Euler-Mascheroni constant and
is the digamma
function.
The only values of 
for which 
is a regular number are 
, 2, and 6 (Havil 2003, pp. 24-25).
The number of terms needed for 
to exceed 1, 2, 3, ... are 1, 4, 11, 31, 83, 227, 616, 1674,
4550, 12367, 33617, 91380, 248397, ... (OEIS A004080;
DeTemple and Wang 1991). Using the analytic form shows that after 
terms, the sum is still less than 20. Furthermore,
to achieve a sum greater than 100, more than 
terms are needed! Written explicitly, the
number of terms is
 |
(8)
|
(Boas and Wrench 1971; Gardner 1984, p. 167). More generally, the number of terms needed to equal or exceed 
, 
, 
, ... are 12367, 15092688622113788323693563264538101449859497,
, ... (OEIS
A096618).
 |
(9)
|
taken over all primes 
also diverges (Wells 1986, p. 41) with asymptotic behavior
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(10)
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(Hardy 1999, p. 50), where 
is the Mertens constant.
Rather surprisingly, the alternating series
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(11)
|
converges to the natural logarithm of 2. An explicit formula for the partial sum of the alternating series is given by
![sum_(k=1)^n((-1)^(k-1))/k=ln2+1/2(-1)^n[psi_0(1/2+1/2n)-psi_0(1+1/2n)].](/images/equations/HarmonicSeries/NumberedEquation8.svg) |
(12)
|
Gardner (1984) notes that this series never reaches an integer sum.
The partial sums of the harmonic series are plotted in the left figure above, together with two related series.
It is not known if the series
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(13)
|
converges (Borwein et al. 2004, p. 56). After 
terms, the series equals approximately 2.163.
