Q: A historian of science is analyzing Newton’s derivation of the binomial series for $(1 + x)^r$. For $|x| < 1$ and real $r$, which identity is central to his expansion?

["Understanding Newton’s Derivation of the Binomial Series for $(1 + x)^r$: The Foundation of the Expansion", "When analyzing Isaac Newton’s groundbreaking work on the binomial series for $(1 + x)^r$, historians of science highlight a pivotal identity that lies at the heart of his derivation. This identity — the infinite power series expansion of $(1 + x)^r$ — enabled Newton to extend traditional algebraic reasoning into the realm of real exponents, fundamentally transforming mathematical analysis.", "Newton’s central insight relied on an intuitive and pioneering approach: representing $(1 + x)^r$ as an infinite sum rather than relying solely on polynomial expansion. Though formal convergence guarantees (such as the condition $|x| < 1$) emerged much later through rigorous analysis, Newton’s derivation hinged on the power of infinite series and iterative reasoning based on the binomial formula:", "$$\n(1 + x)^r = \sum_{n=0}^{\infty} \binom{r}{n} x^n\n$$", "Here, the identity $\binom{r}{n} = \frac{r(r-1)(r-2)\cdots(r - n + 1)}{n!}$, generalized for real $r$, formed the core expression. This generalization of the binomial coefficient allowed Newton to express $(1 + x)^r$ not as a finite term but as an infinite sequence of terms, each involving a factorial and a rising factorial in the numerator.", "Historians emphasize that Newton manipulated these series expansions recursively, validating each step through algebraic consistency and geometric insight. By assuming $r$ is a positive real number and $x$ within convergence, he justified expanding $(1 + x)^r$ term by term, leading to:", "$$\n(1 + x)^r = 1 + rx + \frac{r(r-1)}{2!}x^2 + \frac{r(r-1)(r-2)}{3!}x^3 + \cdots\n$$", "This series not only provided a powerful computational tool but also illuminated deeper connections between algebra, calculus, and infinite processes — laying groundwork for later developments in analysis.", "In sum, the identity $\binom{r}{n} = \frac{r(r-1)\cdots(r-n+1)}{n!}$, generalized for real exponents, stands as the cornerstone of Newton’s derivation. It enabled the shift from finite expansions to infinite series, marking a critical leap in mathematical thought. Recognizing its central role helps historians appreciate both the creativity and rigor behind Newton’s approach to power series and their lasting impact on modern mathematics."]