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How Isaac Newton discovered the binomial power series (2022)
Rethinking questions and chasing patterns led Newton to find the connection between curves and infinite sums.
But along the way he found something even better: a method for replacing complicated curves with infinite sums of simpler building blocks made of powers of$latex x$. Next, to extrapolate his results to half-powers and odd-numbered subscripts (and finally get to the series he wanted, $latex A_1$), Newton needed to extend Pascal’s triangle to a fantastic new regime: halfway in between the rows. That revealed the binomial character of the coefficients in his series — the unexpected appearance of numbers in Pascal’s triangle and their generalizations — which let Newton see patterns that Wallis and others had missed.
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