Appendix Five

We want to show that the harmonic series diverges, in other words that

will add up to infinity. This is done by showing that the harmonic series is larger than the following series, which does add up to infinity:

Let’s compare terms of the harmonic series in groups of two, four, eight and so on, starting from the third term. They are listed below. Because is bigger than must be bigger than which is . Likewise, since and are all bigger than , this means that is bigger than four eighths, which is also . If we carry on, always considering double the number of terms, we can see that we will be able to add up these terms to make a value larger than :

The harmonic series, therefore, is bigger than + + + + + …, which is infinity times a half, which is infinity. So the harmonic series is bigger than infinity; in other words, it is infinite.