Sum of Cubes
What is 13 + 23 + 33 + …+n3 = ?
Glad you asked!
This video answers this specific question using nothing more than a grade-three multiplication table. It is not the most exciting of videos as one should talk about all the different sums of powers (will do .. eventually!) but I do ask some zinger of some questions at the end just to keep us on our toes.
Geometric Series Formula
What is the geometric series formula? When is it valid? How do we derive the formula?
In this video we use a paper-tearing trick to show that the infinite formula is valid at least for some types of numbers (and this trick makes it so obvious that the formula is true in these cases. It’s neat!) We then discuss how to prove the formula true in general and for what range of values it holds. All easy and all fun!
The Basel Problem / The value of zeta two / Wallis' Product
In 1644, Italian mathematician Mengoli asked for the exact value of the infinite sum of the reciprocals of the square numbers:
1 + 1/4 + 1/9 + 1/16 + 1/25 + … = ?
He knew that the sum was bounded and wanted to know its value. Some ninety years later a young Leonhard Euler solved the problem and showed the exact answer to be (pi^2)/6. This is how Euler did it. (We’ll also derive Wallis’s product as a bonus!)
This video is a lead-in to a written essay available on the website that explains Benford’s Law. Here we prove, ever so swiftly(!), that about 5.8% of the powers of two begin with a seven! (And at the end of the video you will be able to prove that just under 10% of those actually begin with 77 and just under 10% of those with 777!)
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