Pi-Squared Over 6: The Algebraic Genius of Euler

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar.  175 problems divided into 35 quizzes with answers at back. Suitable for SAT/Math Contest/Math I/II Subject Tests and Daily/Weekly Problems of the Day. Includes multiple choice, I/II/III case types and constructed response items.
Price is $9.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!

---------------------------------------------------------------
Yes, Pi Day 2010 is "history" unless of course you celebrate July 22nd! Then again, π is so universal in our world that "All π, All The Time" seems appropriate to me. That was a long-winded way of motivating a post about one of the most famous formulas in mathematics:

1/1² + 1/2² + 1/3² + 1/4² + ... + 1/n² + ... = π²/6

The videos below were inspired by one of my most faithful readers, Prof. Jablow. He brilliantly outlined Euler's derivation of the above formula in one of his comments.  I decided to develop it in more detail and provide some background for the younger student. Advanced middle schoolers through undergraduates in college may find this interesting. You might also want to share it with your math team/club.  The 2-part video presumes strong algebraic background and some knowledge of calculus although the latter is not necessary if you simply accept the well-known series expansion for sin(x). You may also find background and details in the excellent Wikipedia article, The Basel Problem.

As always, I add my disclaimer that I am solely responsible for any errors. I know there are a couple of errors in Part II, towards the end. They're pretty obvious and not serious, so I hope they won't ruin it for you! I invite you to comment on these videos both here and on my new YouTube channel, MathNotationsVids.  Of course, as I am finishing this post on 3-25-10 in the AM, YouTubew is down apparently worldwide, so I cannot embed these videos yet!!

Part I of Euler Video



Part II of Euler Video





FURTHER INVESTIGATION
Although the material on infinite series seems quite advanced, middle schoolers can use their graphing or scientific calculators to compute the sum of the first 10, 20, or even 50 terms of the series above. A simple program can also be written on the graphing calculator for summing the first n terms up to, say, n = 500 or 1000. Challenge them to see how "close" they can get to the decimal value of π²/6...
---------------------------------------------------------------------------------------------------------------------

"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)


You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught.
--from South Pacific