18.100B/C FALL 2008: PRACTICE EXAM #2 (1) See problem set 9 for Taylor’s theorem and Taylor series. (2) (a) Let f : X → R be uniformly continuous. Show that if ( a n ) is Cauchy in X , then ( f ( a n )) is convergent in R . (b) Conversely, suppose f : X → R maps Cauchy sequences to convergent sequences. Is it necessarily true that f is uniformly continuous? (3) Give examples of (a) an absolutely convergent series ∑ ∞ n =1 a n with lim sup n →∞ ﬂ ﬂ ﬂ a n +1 a n ﬂ ﬂ ﬂ = ∞ ; (b) a divergents series ∑ ∞ n =1 a n with lim inf n →∞ ﬂ ﬂ ﬂ a n +1 a n ﬂ ﬂ ﬂ = 0 . (4) Assume that ∑ ∞ n =1 a n is a convergent series and that a n ≥ 0 for all n ∈ N . (a) Prove that ∑ ∞ n =1 a 2 n converges. (b) (hard!) Give an example showing that ∑ ∞ n =1 na 2 n does not necessarily converge. (5) (a) Let ( p n ) be a sequence in a metric space ( X, d ), and let a n = d ( p n , p n +1 ). Assume that the series
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Metric space, Limit of a sequence, Cauchy, absolutely convergent series