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Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x)...

Consider the function f defined on R by f(x) = ?0 if x ≤ 0, f(x) = e^(−1/x^2) if x > 0.

Prove that f is indefinitely differentiable on R, and that f(n)(0) = 0 for all n ≥ 1. Conclude that f does not have a converging power series expansion En=0 to ∞[an*x^n] for x near the origin. [Note: This problem illustrates an enormous difference between the notions of real-differentiability and complex-differentiability.]

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