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Chessboard and pyramid

Have you heard of the legend of the chessboard? The story goes like this. There was a wise man who invented the chess game and introduced it to a king. The king loved the game so much that he told the wise man that he could choose anything for a reward. The wise man then pointed to the chessboard and asked for 1 grain of wheat for the first chess square, 2 grains of wheat for the second square, 4 grains of wheat for the third square, 8 grains of wheat for the fourth square, and repeat this doubling pattern. This sounds like a very little reward but at the end the king didn't have enough wheat to reward the wise man.

Today, we will calculate to see how much wheat the wise man actually requested. Is it a lot? Is it little? I'll give you a clue, it's HUGE!

To see how big this reward is, we will calculate the number of pyramids that we could build up using all this wheat.

Sum of reciprocal squares

Today we will look at a very fascinating proof due to Euler for the following identity: $$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \dots = \frac{\pi^2}{6}$$
The mathematician Euler formulated this proof in 1734 when he was 28 year old.

Taylor series

To celebrate the $\pi$ day, in our previous post, we were introduced to a very beautiful identity due to Euler
$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \dots = \frac{\pi^2}{6}$$

The mathematician Euler had an intriguing method to derive this identity. Euler's method employed the Taylor series, so today we will learn about Taylor series, and in the next post, we will look at Euler's technique.