The Harmonic Series

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In mathematics, an infinite series is a sum of an infinite number of items. When summing a finite number of things we always get an finite answer but when we try to sum an infinite number of things we need to be much more careful about how we do it. For example, if we sum the number 1 an infinite number of times clearly the result is not a finite number because the sum just keeps getting bigger and bigger and so we say the sum is infinity. If we sum the number 0 an infinite number of times the result should be the finite number 0. If an infinite sum is finite we say that the series is convergent but if the sum is infinite then the series is divergent.

In this post I would like to consider a particular infinite series called the harmonic series which is named after the harmonics of a pitch in music. The harmonic series provides a good example of the difficulties encountered when we try to sum an infinite number of things. To form the harmonic series we take the following sum:

We would like to determine if the harmonic series is convergent or divergent. It is not immeditely obvious that this sum could diverge because all of the terms are getting smaller and smaller than the previous terms and so we are only adding a smaller number on each time. For example the 100th term of the series is 1/100 = 0.01 and the 1000th term is 1/1000 = 0.001. Judging by this we might guess that the series is going to converge to a finite number because we keep adding only smaller and smaller numbers. However, I warned that we need to be careful when adding an infinite number of terms together.

It turns out that the harmonic series diverges to infinity because the terms are not shrinking fast enough. Our sum just keeps getting bigger and bigger. How can we see that this is true? The key is to group the terms of the series in a clever way. To do this we will add some parenthesis to show how we want to group the terms. We have

Notice that if we add the terms together inside each group of parenthesis then we get a number that is greater than 1/2. Thus the harmonic series is larger than the following sum:

It is clear that adding 1/2 an infinite number of times will give us an infinite sum and since the harmonic series is larger than this sum it follows that the harmonic series diverges to infinity.

Consider the following series:

where we square the terms that occurred in the harmonic series. It turns out that this series converges to a finite number because the terms of the series are now shrinking fast enough because we have squared them and squaring a number less than 1 gives us an even smaller number. The two series we have seen in this post are examples of p series. A p series is a series of the form:

It is a well known fact of mathematics that if p is less than or equal to 1 then the p series diverges and if p > 1 then the series converges. Since the harmonic series is a series with p = 1 it follows that it is divergent as we showed above.

This post has been another adventure in to the land of infinities. Students usually learn about the harmonic series and other p series in their second course on calculus. Here they learn different methods and tests to determine the convergence or divergence of a series including the integral test which uses an improper integral to test for the convergence of a series.

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References:

http://mathworld.wolfram.com/HarmonicSeries.html
https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

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All images were created by myself using latex.