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Pointwise Limit and Uniformity

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Let fk (x) = kxe^-kx, k = 1, 2, 3, . . . .
a) Determine the pointwise limit f of the sequence {fk} infinity k=1 on [0, +infinity).
b) Show that the sequence {fk} infinity k=1 does not converges to f uniformly to f on [0,+infinity).
c) Show that the sequence {fk} infinity k=1 converges to f uniformly to f on [sigma,infinity), where sigma > 0.

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Solution Summary

Determining the point limit of an exponential function sequence. Showing that the sequence does not converge uniformly to f on the interval [0,+infinity), but converges uniformly on [sigma, +infinity), where sigma >0.

Solution provided by:
Chris Busu, PhD (IP)
Education
  • BSc, University of Bucharest
  • MSc, Ovidius
  • MSc, Stony Brook
  • PhD (IP), Stony Brook
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