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I'm supposed to use Lagrange Remainder Theorem to prove that

$$1 + \frac{x}{2} - \frac{x^2}{8} < \sqrt{1+x} < 1 + \frac{x}{2} \text{ } \text{ if } x>0$$

Obviously, the left and right hand sides look like Taylor series, but how would you do this using Lagrange, it isn't obviously intuitive. Thanks!

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    I've added links to mathworld and wikipedia, I hope you don't mind. (This might be useful, since not everybody must know the theorem or someone might know it, but not under this name.)2011-12-02

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