Why Taylor series are more than just clever approximations
In short: Most people see Taylor series as a convenient trick for approximating functions, but there is a rigorous mathematical reason why they actually work. This video explores the Taylor Inequality, which proves that these approximations can be made infinitely precise.
The reliability of a power series depends entirely on its remainder—the difference between the actual function and its polynomial approximation. The Taylor Inequality provides a way to ensure these remainders are 'well controlled.'
For functions like e^x, this mathematical framework guarantees that as you add more terms (n) to the Taylor polynomial, the remainder shrinks toward zero for any value of x. This means that regardless of the required level of precision, there is always a sufficient number of terms available to reach it.
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- Summary by Intellect · source: Why Taylor Series actually work: The Taylor Inequality
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