Limit of Integrals ≠ Integral of Limits | CRAZY Calculus Example!
We explain an important example of a sequence of continuous functions on the interval [0, 1) where limits and integration cannot be interchanged. The sequence {f_n} of continuous functions pointwise converges to a function f(x) (i.e., lim_{n to ∞} f_n(x) = f(x) for each x in [0, 1)) such that:
lim_{n to ∞} ∫ f_n(x) dx ≠ ∫ f(x) dx
We construct the sequence as f_n(x) = nx^n and rigorously prove the above statement is satisfied. The sequence is an important counterexample in real analysis and a great introduction to convergence theorems for integrals (e.g., the Lebesgue dominated convergence theorem). The sequence should also be very interesting to people studying Riemann integration. The video is designed for calculus students with an inclination to theoretical calculus and math proofs, and all real analysis students, especially those interested in Lebesgue integration.
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