Actually, Newton’s series were infinitely more powerful than Ar-chimedes’s in that they enabled him to find the areas of not just one but a whole continuous infinity of circular and hyperbolic segments. That’s what his abstract symbol x did for him. It let him change his problems continuously and effortlessly. It enabled him to tune the shape of segments by sliding x to the left or right so that what appeared to be a single infinite series was in fact an infinite family of infinite series, one for each choice of x. That was the power of power series. They let Newton solve infinitely many
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