SOLUTION: Let {{{f[n](x) = nxe^(-nx^2)}}}, for x in [0,1]. Show whether or not i) {{{f[n](x) }}} converges for each x in [0,1]. ii) {{{f[n](x)}}} converges uniformly on [0,1].

Question 438196: Let , for x in [0,1]. Show whether or not
i) converges for each x in [0,1].
ii) converges uniformly on [0,1].
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
i) Fix any x-value in the interval (0,1].
Now
==>
= by L'Hopital's rule.
=
If x = 0, for all n.
Hence for all x in [0,1], the sequence exhibits pointwise convergence, and .
ii) Now .
Setting this derivative to 0 and solving for x, we get
<==>
Implementing the 1st derivative test using the test points and , we find that there is (absolute) maximum at .
(Remember that .)
Then . Hence as , the maximum of the function goes to infinity, and so the sequence of functions does not exhibit uniform convergence.

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