is called the harmonic series. The harmonic label originates with the
Pythagoreans who believed that a musical string divided by a half, a third, a
quarter, gave the musical notes essential for harmony.
In the harmonic series, smaller and smaller fractions are being added but what
happens to the total? Does it grow beyond all numbers, or is there a barrier
somewhere, a limit that it never rises above? To answer this, the trick is to group
the terms, doubling the runs as we go. If we add the first 8 terms (recognizing
that 8 = 2 × 2 × 2 = 2^3 ) for example
(where S stands for sum) and, because ⅓ is bigger than ¼ and ⅕ is bigger
than ⅛ (and so on), this is greater than
So we can sayand more generallyIf we take k = 20, so that n = 2^20 = 1,048,576 (more than a million terms),
the sum of the series will only have exceeded 11 (see table). It is increasing in an
excruciatingly slow way – but, a value of k can be chosen to make the series total
beyond any preassigned number, however large. The series is said to diverge to
infinity. By contrast, this does not happen with the series of squared terms
We are still using the same process: adding smaller and smaller numbers
together, but this time a limit is reached, and this limit is less than 2. Quite
dramatically the series converges to π^2 /6 = 1.64493...