Infinite Sum of Powers of 1/3 in Equilateral Triangle

Start with an equilateral triangle of area one. The three angle bis sectors meet at the incenter and split the triangle into three congruent triangles, thus dividing the triangle into thirds. The top two shaded triangles represent an area of 2/3. In the remaining unshaded triangle of area 1/3, we can tie the bottom line allowing us to create three equal area triangles as they all have the same base length and same height. If we shade two of these, we have shaded an additional 2/3 of the 1/3 or an additional 29th of the outer triangle. The remaining unshaded triangle is again a new equilateral triangle. So we can repeat the process. First shading 2 over 3 the 3r and then 2 over 3 to 4th. We can repeat this process indefinitely. Each time shading 2 over 3 to the next odd power and then even power and so on. In the limit of this infinite process, the shaded area matches the area of the triangle which is 1. From this visualization, we thus deduce that 2 * the infinite sum of the positive powers of 1/3 is 1. But that means that exactly 1/2.