Why Taylor Series actually work: The Taylor Inequality

A power series for a function is only as good as its remainder. Thankfully, we have an incredibly powerful result for Taylor Series, namely that the remainders are "well controlled" by the Taylor Inequality. In examples like e^x this means that the remainder goes to zero for all values of x as n goes to infinity. That is, no matter how accurate you need me to be, I can take enough terms in my Taylor polynomial and ensure that level of accuracy.

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