For an introduction to this series see this article. Briefly, I show examples of beautiful mathematics that do not need any advanced math.
What is a series?
In mathematics, a sequence is a list of numbers, usually with some defining characteristic. For example the sequence of counting numbers is 1, 2, 3 …. . The sequence of primes is 2, 3, 5, 7, 11….. A series is simply a sequence added up. Several articles in this series (in the non-mathematical sense!) have dealt with series in the mathematical sense. For example, this diary. dealt with the series of consecutive odd integers.
Convergent and divergent series
Some series are said to converge. This means that, no matter how many terms you add, you never get past a certain number. For example the series:
1 + 1/2 + 1/4 + 1/8 + ….. 1/2n + ….
converges to 2. Other series diverge; that is, if you keep adding terms the series grows without limit.
The harmonic series
The harmonic series is the inverse of the sequence of counting numbers:
1/1 + 1/2 + 1/3 + …. + 1/n….
The harmonic series grows very slowly. According to Wolfram’s Mathworld you need to add 12,367 terms to reach 10 and, to reach 100 you need a truly amazing number of terms, specifically: 15,092,688,622,113,788,233,693,563,264,538,101,449,859,497 terms! But does the harmonic series ever stop growing? I will show two proofs that it does. But before reading on I strongly urge you to play! One of the proofs uses only very basic math, but in an ingenious way.
Oresme’s proof that the harmonic series diverges
The first proof that the harmonic series diverges is due to Nicole D’Oresme (c. 1323-1382). He noticed that you could group the terms of the harmonic series like this:
1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16) …
The first parentheses group two terms, the next four and so on. How does this show that the series diverges? Each set of numbers is equal to or greater than 1/2. for example:
- 1/3 + 1/4 > 1/4 + 1/4 and 1/4 + 1/4 = 2/4 = 1/2
- 1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8, which, again = 1/2
Ward’s proof that the harmonic series diverges
A much more recent proof is due to A. J. B Ward in 1970. He used a reductio ad absurdum proof; that is, he assumed the opposite of what he wanted to prove and then deduced nonsense from that assumption. So, first assume that the harmonic series converges. Let Hn be the nth harmonic number e.g. H3 = 1/1 + 1/2 + 1/3. Now, if the harmonic series converges, then as n gets bigger and bigger H2n – Hn approaches 0. (Formally the limit of H2n – Hn as n approaches infinity = 0) . However, Ward noticed that
H2n – Hn = 1/(n+1) + 1/(n+2) + 1/(n+3) +… 1/2n
This series has n terms. Each term is equal to or greater than 1/2n, so the whole series is greater than n/2n which is 1/2. But we assumed that it was 0!