# harmonic sequence

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

1/*a*, 1/(*a* + *d*), 1/(*a* +
2*d*), ..., 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

2*a _{s}a_{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 9^{2}/10.
Similarly, the sum of the terms in (1/100 + ... + 1/999) is at most 9^{3}/10^{2},
etc. So the entire sum is no larger than 9 × 1 + 9 × (9/10) +
9 × (9^{2}/10^{2}) + ... + 9 × (9^{n}/10^{n})
+ ... This is a geometric series that converges. Thus by the comparison
test, the original sum (which is smaller term-by-term) must converge.