A mathematical paradox where the sum of parts equals nothing
In short: Most students assume that if you can calculate every derivative of a function, its Taylor series will eventually reconstruct that function. This video explores a rare class of smooth functions that defy this logic, proving that infinite differentiability does not guarantee analyticity.
The core of the paradox lies in the distinction between 'smooth' and 'analytic.' While most familiar functions are analytic—meaning their Taylor series converges to the original function—certain functions behave differently. The video highlights $e^{-1/x^2}$ (with $f(0)$ defined as 0) as a primary example.
For this specific function, every single derivative at zero is exactly zero. Consequently, its Taylor series is simply a string of zeros. While the series technically converges, it fails to represent the original function anywhere except at the origin. This serves as a vital lesson in calculus: convergence does not always imply equality.
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- Summary by Intellect · source: This Infinitely Differentiable Function Breaks Taylor Series
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