How to Teach High School Students About Harmonic Series When Teachers Can’t Give Answer to a Most Basic Question They Might Ask

Bored students in a mathematics class (Source: Depositphotos by AndreyPopov)

Teaching mathematics subject in High School is not always an enjoyable teaching experience for a lot of reasons. Like any other subject in a classroom setting, several students might not always have a fondness for learning. How much more for a subject that is viewed by many as notorious? Teachers will always find ways to improve on their teaching techniques and strategies but many if not all topics decorated with figures are not just as inviting for fun learning. Perhaps, fun is a mindset but fleeting in reality when figures start to approach the complexity where the devil in the details comes out of infinity, so to speak.

My first teaching experience as an employed teacher in a private school back in 2009 was so memorable, as I grappled not only on mastering my topics but more so on managing the students to learn. It was so immersing, I felt I was like a scorching iron dipped into the water, quenched by the employment that I gained but some effects to my overall well-being like stress must be handled masterfully.

One particular topic that always reverberates in my mind was the lesson on Harmonic Series.

An application of Harmonic Series (Source: YouTube)

Before teaching the lesson on Harmonic Series, topics on Arithmetic and Geometric Series will be taught first which have their formulas completely intact in finding the exact value. Students can explore these concepts and verify the validity of their solution using the formulas given. While the Harmonic Series was left hanging in the air, has no mention of any reason why there are no formulas for both the teachers and students to work on, just exercises using the manual way of computation or iteration to get a particular Harmonic Number, then the textbook concludes in a hanging passion proceeding to the next lesson as if evading a thorough discussion on this.

1/x vs harmonic series area (Source:

Some textbooks or reference materials show a formula for the Sum of Hamonic Sequence or Harmonic Series which are either misleading or completely unverifiable failing any test of reasonableness.

With this in mind, I asked myself why, and that query was also the questions my students had on their pondering mind whose interest made them seeking more from it. Though it became an opportunity for me to arose their mathematical thinking by telling them stories about how mathematicians stumbled on discoveries of mathematical concepts and formulas like one of the famous mathematicians Carl Friedrich Gauss did, a thought that also kept me awake, perhaps for my entire life.

Harmonic Series (Source: Wikipedia)

As I ponder through this Harmonic Series, all it says about were not direct answers to my quest for the formula. It just made mention that it’s a divergent infinite series and that the available formula is just an approximation as discussed by Haris Angelidakis in his article entitled, The Celebrated Harmonic Numbers and How to Approximate Them. Some formulas estimate this Harmonic Progression Sum (another name for Harmonic Series). But still, far from the exact Harmonic Number, we can completely rely on.

The n-th Harmonic Number (Source: Wikipedia)
Approximation of n-th Harmonic Number (Source: Wikipedia)

If this is the case, how could the question about the exact formula for the Harmonic series be handled directly and honestly? Would there be any great significance if this elusive formula will be discovered?

Thank you for reading and sharing your thoughts on this!



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