Harmonic Series

The Sum

$$\sum_{k=1}^\infty {1\over k}$$ (1)

is called the harmonic series. It can be shown to Diverge using the Integral Test by comparison with the function $1/x$. The divergence, however, is very slow. In fact, the sum

$$\sum_{k=1}^\infty {1\over p_k}$$ (2)

taken over all Primes $p_k$ also diverges. The generalization of the harmonic series

$$\zeta(n)\equiv \sum_{k=1}^\infty {1\over k^n}$$ (3)

is known as the Riemann Zeta Function.

The sum of the first few terms of the harmonic series is given analytically by the $n$th Harmonic Number

$$H_n = \sum_{j=1}^n {1\over j} = \gamma+\psi_0(n+1),$$ (4)

where $\gamma$ is the Euler-Mascheroni Constant and $\Psi(x)=\psi_0(x)$ is the Digamma Function. The number of terms needed to exceed 1, 2, 3, ... are 1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, ... (Sloane's A004080). Using the analytic form shows that after $2.5\times 10^8$ terms, the sum is still less than 20. Furthermore, to achieve a sum greater than 100, more than $1.509\times 10^{43}$ terms are needed!

Progressions of the form

$${1\over a_1}, {1\over a_1+d}, {1\over a_1+2d}, \dots$$ (5)

are also sometimes called harmonic series (Beyer 1987).

See also Arithmetic Series, Bernoulli's Paradox, Book Stacking Problem, Euler Sum, Zipf's Law

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 279-280, 1985.

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 8, 1987.

Boas, R. P. and Wrench, J. W. ``Partial Sums of the Harmonic Series.'' Amer. Math. Monthly 78, 864-870, 1971.

Honsberger, R. ``An Intriguing Series.'' Ch. 10 in Mathematical Gems II. Washington, DC: Math. Assoc. Amer., pp. 98-103, 1976.

Sloane, N. J. A. Sequence A004080 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.