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A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n << 1$. Does this approximation have a name? Any leads on estimating the error of the approximation?

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First order Taylor approximation or linearization. – Grumpy Parsnip Oct 9 '12 at 22:45
Taylor approximation is exactly what I was looking for. Somehow, given that (1+x)^n has a finite expansion, I thought this was about finite series rather than infinite series, and didn't even think of the Taylor expansion. Your comment made me realize that the finite expansion IS the Taylor series, so I can use the integral form of the remainder to estimate how good of an approximation we have. Thanks! – Martin C. Martin Oct 10 '12 at 18:40

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I would just call it the first order truncation of the Binomial series. If you want more terms of the series, then it's given by $$(1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \mathcal{O}(x^4)$$ for the full series, you can visit the link I provided.

You may also be interested in Bernoulli's inequality

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I would say it comes from the Bernoulli inequality. You can read it up on Wiki

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