Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 2G / b / iv

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- Solution iv

Solution

The pointwise limit is zero, including at

x = 0

. Differentiation gives

\frac{d}{d x} (x e^{- n x}) = e^{- n x} (1 - n x),

so the maximum on

[0, \infty)

is attained at

x = 1/ n

and equals

1/ (e n)

. Therefore

x \geq 0 sup x e^{- n x} = \frac{1}{e n} \to 0,

and the convergence is uniform, even though the domain is unbounded.

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