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PROBLEM SET 11 (DUE IN LECTURE ON DEC 4 (FRIDAY)) (All Theorem and Exercise numbers are references to the textbook by Apostol; for instance “Exercise 1.15-3” means Exercise 3 in section 1.15.) Problem 1. Let f : R → R be defined by ( e−1/x for x > 0 f= . 0 for x ≤ 0 (a) Prove that f is continuous at 0. (b) Prove that f is differentiable at 0. (c) Determine whether or not the derivative of f is continuous at 0. (Hint for all parts: you might find it useful to transform the limits in question into limits as x → +∞ by replacing x by x1 .) Problem 2. Do Exercise 10.4-7. Problem 3. Do Exercise 10.9-3 Problem 4. Compute the sum of the series ∞ X π 4n . (4n)! n=0 (Hint: use the power series formulas for exp and cos that were proved in class (and can be found in section 11.11 of Apostol).) Problem 5. Define a sequence {an } by a1 = 2 and an+1 = an + 2 an 2 for n ∈ N. The first few terms of the sequence are: 3 17 577 665857 2, , , , ,... 2 12 408 470832 Computing decimal expansions of these fractions √ will quickly convince you that the sequence is converging very rapidly to 2 = 1.41421356237310 . . .; for instance 665857 a5 = = 1.41421356237469 . . . 470832 is already correct to many decimal places. The √ goal of this problem is to prove that this sequence √ does in fact converge to 2.2 √ (a) Prove that an ≥ 2 for all n by showing that x+ x ≥ 2 2 for any positive √ real x. (Hint: find the minimum of x + x2 on the interval [ 2√1 2 , 2 2].) (b) Use the previous part to show that the sequence {an } is decreasing. (c) Conclude that √ the sequence must converge (why?) and prove that it converges to 2. 1