Euler: The Intro Continues

Posted: February 27, 2016 in Calculus: An Introduction
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There are many great explorations to pursue in Math 31 (introduction to Calculus) of which we owe thanks to Leibniz, Newton and many others who followed. With “π” week fast approaching, it is an appropriate time to further introduce my students to the genius of Swiss mathematician Leonhard Euler.

This entry begins with a step function that appears to converge on π. Through our little journey here, students will understand and develop the reasoning used by Euler to show that the infinite series 1/n^2 does in fact converge.

Euler pi

What’s going on here and why is this thing approaching π???

 

To have any chance of understanding this, we need to build on what students already know. The following set-up is crucial.

(pi)squaredBY6

The notes directly above are well within the abilities of high school students to conceptualize; these same concepts and procedures will now be transferred to a new context that students are not so familiar with. The Taylor Series allows us to approximate many functions very accurately (within the vicinity of some value of “x”) as polynomial functions. The polynomial representation of f(x)=sin(x) will be exploited here to arrive at our destination.

 

Taylor Series

(pi)squaredBY6(b)

sin(x)

 

Students are well-versed in expressing polynomial functions in factored form as shown below; an appropriate comparison is eventually made to end our little journey.

(pi)squaredBY6(c)

Thanks for reading.

 

Reference

Courant, Richard., John, Fritz (1999).  Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY:  Springer-Verlag Berlin Heidelberg.

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