Using the natural numbers, we can construct a series by using the reciprocal of each natural number. If we do so, we get the series 1/1 + ½ + 1/3 + ¼ + 1/5 + ... This series is called the harmonic series, a series that has been studied since ancient times.
Here are the values of the sum of the first n terms of the harmonic series:
|n||Sum of first n terms of Harmonic Series (approximately)|
An interesting question to ask is whether the harmonic series converges (approaches a finite value as the number of terms in the series approaches infinity) or diverges (approaches infinity as the number of terms in the series approaches infinity). Looking at the series and the table above, it might appear that the series converges, because the terms keep getting smaller and smaller and the series grows at increasingly smaller rates. However, this is not the case.
Nicole Oresme (ca. 1323–1382), the Bishop of Lisieux, France, proved that the harmonic series diverged and was therefore unbounded. He did this in a surprisingly simple way. Here's the harmonic series again:
We can group the terms as follows:
In each bracketed group, we'll replace each fraction with the smallest fraction in that subset. This substitution will yield a new series with a sum that is smaller than the harmonic series. So, we get:
Adding the terms together in each subset, we get:
Since there are an infinite number of subsets, there is also an infinite number of ½'s to be added, so the result will be infinite. Since this smaller series diverges, the harmonic series diverges as well.
It is interesting to note, however, that 1⁄1n + 1⁄2n + 1⁄3n + 1⁄4n + ... converges to a finite sum if n is any real number greater than 1, so in a certain sense we could say that the harmonic series "just barely" diverges.