Harmonic series (mathematics)

In mathematics, the harmonic series is the divergent infinite series:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ... = ∑_(n=1)^∞ 1/n

More generally, any series of the form

∑_(n=1)^∞ 1/(an+b)

where a and b are constants, and a is non-zero, is also called a harmonic series.

The harmonic series is a classic example of a series that diverges even though its individual terms approach zero. This divergence can be demonstrated in several ways. One common method involves grouping terms and showing that each group sums to a value greater than a constant. For instance:

1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ...

Here, 1/3 + 1/4 > 1/4 + 1/4 = 1/2; 1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8 = 1/2; and so on. Since each grouped term is greater than 1/2, and there are infinitely many such groups, the series diverges.

Another way to demonstrate divergence involves comparing the harmonic series to the integral of 1/x. The integral of 1/x from 1 to infinity is infinite, and the harmonic series represents a Riemann sum approximation of this integral. Because the function 1/x is decreasing, the harmonic series overestimates the integral, thus implying divergence.

Despite its divergence, the harmonic series grows extremely slowly. The partial sums of the harmonic series, denoted H_n = ∑_(k=1)^n 1/k, are closely related to the natural logarithm. Specifically, H_n ≈ ln(n) + γ, where γ is the Euler–Mascheroni constant (approximately 0.57721). This relationship allows for approximations of the partial sums and is crucial in various applications.

Modified versions of the harmonic series, such as the alternating harmonic series (1 - 1/2 + 1/3 - 1/4 + ...), converge conditionally. In general, a p-series ∑_(n=1)^∞ 1/n^p converges if p > 1 and diverges if p ≤ 1. The harmonic series is the special case where p = 1.