A harmonic sequence, also called a harmonic progression, is a sequence of the form:

 

    1/a, 1/(a + d), 1/(a + 2d), ..., 1(a + (n – 1)d),

 

the terms being the reciprocals of those in an arithmetic sequence. There is no simple expression for the sum of a harmonic progression. The harmonic mean of two terms, a_s and a_x+2 is given by

 

    2a_sa_s+2 / (a_s + a_s+2 ) = a_s+1.

 

Consider the sequence: 1, 1/2, 1/3, 1/4, 1/5... Added together, these become the terms of the harmonic series: 1 + 1/2 + 1/3 + 1/4 + 1/5 +... This series diverges (has no finite sum), though very slowly – a result first proved by the French philosopher and theologian, Nicole Oresme (c.1325–1382). In fact, it still diverges if you take away every other term, and even if you take away nine out of every ten terms. However, if you take the sum of reciprocals of all natural numbers that do not contain the number nine (when written in decimal expansion) the series converges! To show this, group the terms based on the number of digits in their denominator. There are 8 terms in (1/1 + ... + 1/8), each of which is no larger than 1. Consider the next group (1/10 + ... + 1/88). The number of terms is at most the number of ways to choose two ordered digits out of the digits 0 ... 8, and each such term is clearly no larger than 1/10. So this group's sum is no larger than 9²/10. Similarly, the sum of the terms in (1/100 + ... + 1/999) is at most 9³/10², etc. So the entire sum is no larger than 9 × 1 + 9 × (9/10) + 9 × (9²/10²) + ... + 9 × (9ⁿ/10ⁿ) + ... This is a geometric series that converges. Thus by the comparison test, the original sum (which is smaller term-by-term) must converge.