Proofs Without Words

$$1+\frac{1}{4}+\frac{1}{9}+\cdots = \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi ^2}{6}$$

Here's an elementary way to find the area of a circle although I'm sure many of you are already familiar with it. Slice up the circle into 4 parts and stack them up as follows.

What is the Sierpinski Triangle? Let's consider an equilateral triangle. Now remove the equilateral triangle joining the mid-points of the sides of the triangle. Notice that the area of the new figure is $\frac{3}{4}^{\text{th}}$ the initial area. Now lets apply the same procedure to the 3 smaller triangles.

$1+2^2+3^2+\cdots+n^2 = \frac{1}{3}n(n+1)\left(n+\frac{1}{2}\right)=\frac{n(n+1)(2n+1)}{6}$ This originally came in Mathematics Magazine (1984 edition). The author of this ingenious proof is Prof. Siu Man Keung. This link provides further information regarding the article.