How Isaac Newton Discovered the Binomial Power Series (2022)

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Isaac Newton discovered the binomial power series through a process of analogy and pattern recognition.
Newton was inspired by John Wallis's work and sought to find the area under a circular segment defined by y = √(1 - x²).
He examined areas under curves with whole-number powers first, which were easier to calculate, and then tried to generalize to half-powers.
Newton noticed patterns in the coefficients of the series, linking them to binomial coefficients and Pascal's triangle.
By extrapolating these patterns, he derived a general formula for binomial coefficients, even for fractional exponents.
This led him to a power series representation for the area under the circular segment and, by extension, an infinite series for π/4.
Newton realized that the circle's equation itself could be represented as a power series, expanding the utility of his method.
His approach demonstrated the power of generalization and pattern recognition in solving mathematical problems.
Newton's binomial power series became a foundational tool in calculus, enabling solutions to a wide range of problems.

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