Unformatted text preview: R be a continuous function. If lim x →∞ f ( x ) exists and is ﬁnite, then prove that f is actually uniformly continuous. (Note that Theorem 19.2 does not apply because the domain of f is not bounded.) Problem 5: Determine the radius of convergence of the series ∑ ∞ n =0 a n x n , where: a n = 5 n , if n is odd 1 2 n , if n is even Problem 6: Deﬁne functions f n : R → R by f n ( x ) = cos n x . Note that these functions are all continuous. Prove that: (a) lim n →∞ f n ( x ) = 0 if x is not a multiple of π . (b) lim n →∞ f n ( x ) = 1 if x is an even multiple of π . (c) lim n →∞ f n ( x ) does not exist if x is an odd multiple of π ....
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- Fall '07
- Calculus, lim, Continuous function, lim fn