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UNIVERSITY OF DUBLIN XMA1213 TRINITY COLLEGE Faculty of Science school of mathematics Hilary Term 2008 JF Maths, JF TP JF TSM, SF TSM Course 121 Monday, March 10 Regent House 14:00 – 17:00 Dr. P. Karageorgis Attempt all questions. All questions are weighted equally. You may use non-programmable calculators, but you may not use log tables. Page 2 of 2 XMA1213 1. Compute each of the following integrals: Z 3x − 1 dx, x3 − x Z x log x dx. 2. Suppose f, g are integrable on [a, b] with f (x) ≤ g(x) for all x ∈ [a, b]. Show that Z b Z b f (x) dx ≤ g(x) dx. a a 3. Define a sequence {an } by setting a1 = 1 and an+1 = √ 3an − 1 for each n ≥ 1. Show that 1 ≤ an ≤ an+1 ≤ 3 for each n ≥ 1, use this fact to conclude that the sequence converges and then find its limit. 4. Compute each of the following limits: lim x→1 x3 − 5x2 + 7x − 3 , x3 − 4x2 + 5x − 2 lim x sin(1/x). x→∞ 5. Test each of the following series for convergence: ∞ X (−1)n−1 e1/n n=1 n ∞ X µ 1 log 1 + n n=1 , ¶ . 6. Find the radius of convergence of the power series f (x) = ∞ X (n!)2 n ·x . (2n)! n=0 7. Suppose f is a differentiable function such that f 0 (x) = f (x) + ex for all x ∈ R. Show that there exists some constant C such that f (x) = xex + Cex for all x ∈ R. 8. Use the formula for a geometric series to show that ∞ X n=0 n 2 xn = x(1 + x) (1 − x)3 whenever |x| < 1. c UNIVERSITY OF DUBLIN 2008 °