A Wikipedia-Style Proof

We wish to show that for small $x$, $$\sqrt{1+x} \approx 1 + \frac{x}{2}$$ To derive this formula, we use the well known fact that $$(1+x)^\alpha = \sum_{n=0}^\infty \frac{x^n}{n!} \prod_{k=0}^{n-1} (\alpha - k)$$ The result follows by substituting $\alpha = ½$ and truncating the series after two terms.

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