A famous formula for π: π/4 = 1 − 1/3 + 1/5 − 1/7 + … This comes from a trigonometry formula: arctan(x) = x − x³/3 + x⁵/5 − x⁷/7 + … Putting x = 1 gives the series for π/4. This result was discovered independently by: Gottfried Wilhelm Leibniz (1673), James Gregory (1671), Nilakantha Somayaji (around 1500, in Tantrasangraha).

Infinite Geometric Series Visualization A geometric proof that 1/3 = ∑_{i=1}^∞ 1/4^i = 1/4 + 1/4² + 1/4³ + 1/4⁴ + 1/4⁵ + ⋯ The shaded triangles inside the large triangle represent the successive terms of the series; their areas sum to exactly one-third the area of the large triangle.

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According to The Inquisitive Problem Solver, the only positive integer solution to: A × B × C = C × D × E = E × F × G is: 8 × 1 × 9 = 9 × 2 × 4 = 4 × 6 × 3, if we assume that each variable must be a single digit.

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