# Limiting the error using Taylor's formula with Lagrange's Remainder

I get stuck on problems about approximating values with Taylor's formula when it comes to determining the amount of taylor terms (n) required to satisfy a minumum error by just using Lagrange's remainder.

Example: Approximate $\sqrt[3]6$ (cube root) with an error that is less than $10^{-4}$.

I typically try to use the remainderterm and inequalities to determine n, but my success is infrequent so I'm not sure I do things right.

Any help is appreciated.

Using your example : you can write $$\sqrt[3]6=\sqrt[3]{8-2}=2\sqrt[3]{1-\frac{1}{4}}$$ Now, consider the Taylor expansion of $(1-x)^{1/3}$ built at $x=0$ or use the binomial theorem to get $$2(1-x)^{1/3}=2-\frac{2 x}{3}-\frac{2 x^2}{9}-\frac{10 x^3}{81}-\frac{20 x^4}{243}-\frac{44 x^5}{729}-\frac{308 x^6}{6561}+O\left(x^7\right)$$ where $x=\frac{1}{4}$