There are several instances instances of mathematical mathematical systems defying intuition. intuition. Some of the most intriguing mathematical mathematical anomalies are telescoping telescoping series. These series, which are are the summation of an infinite set of numbers, end up cancelling out nearly all of their terms in a twist of mathematical mathematical genius. There is no specific setup setup for a telescoping series; it is merely allowed to do this because of its property of being a partial sum. For example, take the series defined in Formula 1:
∑ ( )
If we use partial sums this simplifies to:
) ∑ (
This series goes from one to infinity, so we can write out a few terms and see if there is a dominant pattern:
Almost all of the terms cancel with the preceding or following one. The last terms remaining are being evaluated as n→∞, so the final term is going to zero. This leads to the deduction that the entire, infinite sum of numbers converges to one due to the repeated cancellation of subsequent terms. These examples of a cancelling sum often arise in mathematics, and are many times the basis of many people’s intrigue due to the perplexity of cancelling an infinite amount of terms. Another point of intrigue in common Calculus is the harmonic series. This derives from
the function of . Dr. Julian Fleron of Westfield State College describes this function in his paper, detailing the profound properties of it as it is revolved around the x-axis. He simplifies
the graph by making the line a step function. That is, for each interval of one, there is a
horizontal line representing a general estimate of . Once revolved, this comes to look like a tiered cake, hence Fleron’s identifying of a wedding cake. The properties become interesting when we take into account the volume and area of the new three-dimensional cone. To find the volume of the solid of revolution we just sum the two-dimensional volume of
:
∑ ()
We can see using our series tests that this is convergent due to the p-series test. Therefore, this “cake” has a finite volume although it goes on to infinity in the x direction: very counterintuitive. Secondly, the surface area of this figure can be demonstrated through a series like this:
∑ ()
This, resembling the original function, is a classic harmonic series. Again using the p-series test, we can conclude that this series indeed diverges; so our figure has an infinite surface area. Due to our knowledge of solids of revolutions and series tests we can conclude that this infinite threedimensional object is quite curious. The extrusion does have a resemblance to a cake, as seen below, but remember; it goes on to infinity [3].
Figure 1: Gabriel's Wedding Cake
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This representation of the harmonic series is only one of many that show its immense flexibility. This series has many practical applications due to its simplicity and explicitness. One of the most practical uses is that it shows very simply that general convergence of the terms to zero does not imply convergence of the series. The terms in
slowly go towards zero, yet the
series diverges, as stated before. No other series can prove this paradox as simply. Secondly, many instances of the comparison tests in series evaluation would be more difficult if not for the harmonic series, for it is a short and sweet function that diverges and therefore can be used to prove other series’ divergence. The harmonic series is simply the most used and valuable infinite series, and has been around for a great time, dating back to Nicolas Oresme in the 14
th
century [2]. The celebration of this series is not rare, for even Dr. Feland himself admitted that it is “the most important of all the infinite series” [1]. In conclusion, Gabriel’s Wedding Cake is a structural anomaly that truly displays the complexity and intrigue of mathematics; the concept itself is absolutely amazing. For example, say this cake was a tangible object with an infinite area but volume of
, as it is accepted to be.
If the scale was set so that each unit was, say, one hydrogen molecule, we would have a very interesting case on our hands. If you took every hydrogen atom in the universe, you could still not cover the outside of it. Yet you couldn’t even fit one whole atom inside it, specifically only about 0.175. This paradox is mind-blowing, yet completely proven by fairly simple mathematics. This only strengthens the awareness of the immense manipulatory and explanatory powers of math, yet it only scratches the surface of its extensiveness.
1. 2. 3.
Fleron, Julian F., Gabriel’s Wedding Cake, v 30. P 35-38. Hersey, George L., Architecture and Geometry in the Age of the Baroque, p 11-12 and p37-51. Kifowit, Steve and Stamps, Terra, Serious About the Harmonic Series , November 10, 2005.
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