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Limits and uniform convergence

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Consider the sequence of functions f_n(x) =sin ([2npi)^2+x]^(1/2)) on [0,infinity)

a. Show that lim( n goes to infinity) f_n(x)=0
Hint: Use the mean value theorem for f_n(x) on [0,x]

b. Show that f_n(x) converges uniformly to 0 on [0,a] for a fixed a in [0,infinity).

c. Is it true that f_n(x) converges uniformly to 0 on [0, infinity)? Justify your answer.

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Solution Preview

Proof:
on
Then we have

(a) I claim that .
We fixed an . According to the Mean Value Theorem, we have
, where ...

Solution Summary

Limits and uniform convergence are reiterated.

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