Midterm Examination – Math 142B
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Midterm Examination – Math 142B
Question 1. Let (an )n≥1 be a sequence of real numbers. In this question you may use without proof the inequalities 2x/π ≤ sin x ≤ x for x ∈ [0, 1].
(a) State Cauchy’s criterion for convergence of the series 对 an . (b) Prove that if an = sin(1/n) for n ≥ 1, then 对 an diverges.
(c) Prove that if an = (−1)n sin(1/n) for n ≥ 1, then 对 an converges.
(d) Prove that if an = (1/n)sin(1/n) for n ≥ 1, then 对 an converges.
Question 2. For n ≥ 1 and x ∈ R, let
fn (x) = 
and f(x) = 
fn (x).
n
Let r ∈ [0, 1) and sn (r) = sup{ 'f(x) −工 fn (x)' : −1 ≤ x ≤ r}.
k=1
(a) Prove that sn (r) → 0 as n → ∞ .
(b) Prove that 对 fn (x) → f(x) uniformly on [−1,r].
(c) Prove that 对 fn (x) does not converge uniformly to f(x) on (−1, 1).
(d) Is f continuous at x = −1?
Question 3. Let f(x) = (sinx)2 for x ∈ R and let [x] denote the largest integer less than or equal to x.
(a) State Taylor’s Theorem.
(b) Prove by mathematical induction on n that for n ≥ 1,
f (n)(x) = 
(c) Write down the Taylor series for f(x) about x = 0.
(d) Determine for which x ∈ R the Taylor series for f(x) equals f(x).
2022-09-01