New Pi Formula, Baran Binomal

The derivation is elementary and self-contained. The starting point is the well-known geometric identity that the area of the unit quarter-circle equals π/4, expressed as the definite integral of √(1−x²) over [0, 1]. Newton's generalized binomial series is then applied to expand √(1−x²) as a power series in x, and term-by-term integration over [0, 1] yields a series for π/4. After substituting the closed form of the binomial coefficients C(1/2, k) in terms of factorials and simplifying the resulting expression algebraically, the formula above is obtained. The series converges at rate O(k^{−5/2}), meaning the k-th term behaves asymptotically as 1/(4√π · k^{5/2}). While the convergence is slow compared to modern computational formulas such as Machin-type arctangent series, it is faster than the classical Leibniz–Gregory series (which converges at rate O(k^{−1})). The primary value of this result lies in its clean derivation directly from the geometry of the circle and a single application of the binomial theorem, without appeal to trigonometric identities or calculus beyond basic integration. Keywords: pi, infinite series, binomial series, unit circle, slowly converging series, factorial series.