Mathematics · Sequence of Functions
For each n ≥ 1, let fn: R → R be a function defined by fn(x) = e-n2x2n. Which of the following statements is true?
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- 1(fn)n≥1 converges pointwise to 0 on R but not uniformly on R.
- 2(fn)n≥1 converges uniformly to 0 on R, and (f'n)n≥1 does not converge pointwise to 0 on R.
- 3(fn)n≥1 converges uniformly to 0 on R, and (f'n)n≥1 converges pointwise to 0 on R.✓
- 4(fn)n≥1 converges uniformly to 0 on R, and (f'n)n≥1 converges uniformly to 0 on the interval (-M, M) for some positive real number M.
Correct answer
(fn)n≥1 converges uniformly to 0 on R, and (f'n)n≥1 converges pointwise to 0 on R.
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