Sum of Inverse Powers of Three

$\sum_{n=1}^\infty \frac{1}{3^n} = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots = \frac{1}{2} $


The square gets divided into three parts. One of them is marked black (its area is $\frac{1}{3}$), and then in the second one of the three parts we recursively repeat this division process. The area of the black parts is then exactly $\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots$. Hover over the visualization to see a rearrangement of the black parts into a triangle of area $\frac{1}{2}$.


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