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LANCASTER UNIVERSITY
2009 EXAMINATIONS
PART II (Third or Fourth Year) MATHEMATICS & STATISTICS 2 Hours Math 314: Integration
You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A
A1. A word is a finite string of letters from the Roman alphabet. (i) Give an upper bound on the number of words with n letters. (ii) Show that the set of all words is infinite, but countable. [8]
A2. (i) State Archimedes’ Axiom. (ii) Deduce that any irrational number is the limit of some sequence of rational numbers, whereas any rational number is the limit of some sequence of irrational numbers. (iii) Let f be the comb function f (x) =
{ (^1) , 0 < x < 1 and x rational, 0 , else. Show that f is discontinuous at all points of (0, 1), and determine with justification the value of the Lebesgue integral (^) ∫ (^1) 0 f^ (x)^ dx.^
[13]
A3. By using integration by parts, or otherwise, show that ∫ (^) ∞ 0 sin^ x e
−txdx = 1 1 + t^2 (t >^ 0).^
[10]
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SECTION A continued
A4. (i) State how the outer measure of a subset F of R is defined. (ii) Show from first principles that Z has outer measure zero. [10]
A5. State the main theorem (Σ)^ concerning the properties of Lebesgue measure m. [9]
SECTION B
B1. (i) With the aid of a suitable diagram, determine the sets {y ∈ R : y > 1 , sin y > λ} and {x ∈ (0, 1) : sin (^1) x > λ} for 0 < λ < 1; explain why the sets are measurable. [8] (ii) Let f : (a, b) → R be a continuous function. Prove in detail that f is measurable. [6] (iii) Show that g is an integrable function on (1, ∞), where g(x) = sin x 2 x (x > 1). (^) [6]
(iv) Show that h is not integrable on (1, ∞) where h(x) = sin x^ x (x > 1). [10]
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SECTION B continued
B3. (i) State the Dominated Convergence Theorem. [8] (ii) Using power series or otherwise, show that eu^ ≤ 1 /(1 − u) for 0 < u < 1. [4] (iii) Deduce that 1 − u ≤ e−u^ for 0 < u < 1, and hence that (1 − x^2 /n)n^ ≤ e−x^2 (0 < x ≤ √n). [4]
(iv) Deduce that (^) ∫ √n 0 (1^ −^ x
(^2) /n)ndx →^ ∫^ ∞ 0 e
−x^2 dx as n → ∞. [8]
(v) Express the left-hand side of (iii) as a trigonometric integral by substituting x = √n sin t. [6]
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