A New Preception of Harmonic Series

By: Gaurav Kumar (M.Sc. Mathematics)

How I come up to the idea of writing this blog:
During my 12th grade class, I come up with a very painful expression in maths a nightmare symbol
which makes every single question so difficult to solve and also this idea that how can we write
left side which is summation take discrete values were as our right side is a continuous function.
I was even more confused when is see graphical plotting of this continuous function of summation, like what is going on in negative side.
Figure 1: Harmonic Series
History of Harmonic Series
1. Many mathematicians have contributed to the study of series. For example, Carl Friedrich Gauss was a remarkably intelligent mathematician who demonstrated his skill in class one day. His teacher, as a form of punishment, assigned the students the task of finding the sum of the numbers from 1 to 100. While this seemed like it would take a long time, Gauss completed it almost instantly by using a simple and logical approach. What he did was...
  as adding both give
  It’s especially interesting to note that Gauss was just a child when he achieved this! As he and other mathematicians, like Leonhard Euler, continued their work, they expanded their studies on various series, including the harmonic series. In his research, Euler developed an equation involving H(x), representing the harmonic series
which is the approximation of values of harmonic series and it is nothing but a shifted copy of Digamma function and the most fascinating part is that this shift involves a mysterious constant that appears in various areas of mathematics and physics. Known as the Euler-Mascheroni constant, this number emerges in real-world applications, such as quantum corrections to the mass of the electron, the Higgs boson, and even in corrections to the charge of the electron and it is represented as γ
Figure 2: Euler-Mascheroni constant
The light brown region in the graph represents the Euler-Mascheroni constant, whose properties remain somewhat mysterious, we still don’t know whether it is rational, irrational, or transcendental. As shown in the graph, we can approximate H(x) by integrating 1/x from 1 to infinity, and adding light brown region essentially reflects the constant term that, when added to the integral, allows us to calculate H(x) for any x.

If we add up all the remaining area from the above graph other than area under 1/x we get that Euler-Mascheroni constant which is approximately 0.557...
This formula can be extended to all real and that is why we see all this negative side in graph and proof of this expression is so big that this blog will be of 10-15 pages, it’s make our blog boring.
Nowadays, the harmonic series seems to pop up everywhere, even in physics problems. Take, for example, the ”Leaning Tower of Lire” problem posed by Paul B. Johnson. In this problem, you’re given an infinite number of identical coins and asked to stack them over an edge to maximize their overhang, or the horizontal distance from the edge of a support surface, without causing them to topple. Interestingly, this setup involves the concept of the harmonic series, as each additional coin increases the overhang by a fraction that follows a harmonic progression. The resulting overhang pattern looks something like this:
Figure 3: Leaning Tower of Lire
Interesting part in this Harmonic Series is divergent so theoretically in favorable condition when can get infinite overhang but for that you need to have so much Lire which are kind of expensive, and proof of having the most overhand with only one coin in each row is Harmonic Series is just done recently by David Treeby in it’s ”Further thoughts on a Paradoxical Tower”. Harmonic Series is a divergent series and its proof goes back to approximately 700 years by ”Nicole Oresme” this divergent series is consider big daddy of divergent series as it contains lot of divergent series with in itself which makes it very powerful series with it’s wide range of application in Divergence and Error Analysis in Series and Approximations, like it contain Geometric Series of power of 1/2 like
as Geometric Series of power of 1/2 converges to 2, here is a Quick Question does all the Odd and Even detonation only entries series converges or diverges? like
I have digged this topic so much in last few months and i want to talk all day long over it and want everyone to know about it’s useful applications and proofs but this topic and my urge to explain over it is same level as it’s divergent nature of our series so it’s better to finish this introduction and finish this blog with some real life application of Harmonic Series and see you all in next one.
Application of Harmonic series
1. In Number Theory and Prime Analysis as it helps in estimating the density of primes and provides approximations for functions that count prime numbers.
2. Computer Science and Algorithm Analysis like in finding average time complexity of algorithms like Quick Sort and Binary Search Tree as it accounts for cumulative cost of operations.
3. In Physics and for frequency and resonance of musical instruments as it has its own theory when you study musical instrument, and also in all the application where waves are involved
4. In Economics and Finances as series can model the growth of interest over time when compounding it not continuous.
5. Most important is it’s application in Riemann Zeta function.
  
  There are so many application of this divergent series in our real life and how it help us understand phenomenons from damping oscillation to a million problem of Riemann Zeta function and, understanding such a crucial concept of mathematics is a necessary skill in today’s world.