harmonic series | Physics Library

The harmonic series

∑∞ 1 1 1 k-= 1 + 2-+ 3-+ ... k=1

satisfies the necessary condition of convergence

lim ak = 0 k→ ∞

for a series

a1 + a2 + a3 + ...

of real or complex terms. In particular,

1- kli→m∞ k = 0.

Nevertheless, the harmonic series diverges. This can be seen by grouping the terms as

( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 + 2-+ 3-+ 4- + 5-+ 6-+ 7-+ 8- + 9-+ 10-+ ...+ 16- + ...

Here, each parenthesized sum contains twice as many terms as the preceding one. The sum in the first parentheses is greater than

1 1 2 ⋅--= --, 4 2

and the sum in the second parentheses is greater than

1- 1- 4 ⋅ 8 = 2 .

Continuing in this way, one sees that every parenthesized block is greater than . Consequently, the partial sum of the first n terms eventually exceeds any given real number, and therefore the harmonic series diverges.

The divergence of the harmonic series is very slow, though. Its speed may be illustrated by considering the difference

n−1 ∫ n−1 ∑ 1- n dx- ∑ 1- k − x = k − ln n k=1 1 k=1

(see the diagram). We know that ln n increases very slowly as n →∞ (for example, ln(1 000 000 000) ≈ 20.7). The increase of the partial sum ∑ _k=1ⁿ⁻¹ 1
k is about the same, since the limit

( ) n∑−1 1 nl→im∞ --− lnn = γ k=1 k

is a small positive number,

γ = 0.5772156649 ...,

called the Euler constant or Euler–Mascheroni constant.

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