| 61. Prove 1 + x + x4 + x9 + ... + xn2 + ... → ∞ as x → 1. How fast? [In other words, find upper and lower bounds for xn.] | |
| 62. Show that the polynomial xn + x = 1 has a unique positive root kn. Show that kn → 1 as n → ∞. How fast? | |
| 63. Define y1 = 1, yn = sin yn-1 for n > 1. Prove yn → 0 as n → ∞. How fast? | |
| 64. Define f(x) = 1/(1 + x) + 1/(2 + x) + 1/(4 + x) + ... + 1/(2n + x) + ... . Prove f(x) → 0 as x → ∞. How fast? | |
| 65. Let U be a set with n elements. Let K be a set of subsets of U, each with three elements, such that the intersection of any two distinct elements of K is either empty or contains just one element. Let f(n) be the largest possible number of elements that K can have. Estimate f(n). [In other words, find upper and lower bounds for f(n).] | |
| 66. Let S be the sequence of positive integers divisible only by 2 or 3 arranged in ascending order. [The sequence starts: 1, 2, 3, 4, 6, 8, ... ] Prove that the ratio of successive terms tends to 1. | |
| 67. We say that the lattice point P is visible from the origin O if the segment OP does not contain any other lattice points. Show that the proportion of lattice points in the square 0 < x, y ≤ n which are visible from the origin tends to a limit and find it. [Lattice points are points in the Euclidean plane with integral coordinates.] | |
| 68. Let f(k) be the number of lattice points in the disk radius k centered on the origin. Prove that f(k) = πk2 + O(k) as k → ∞. | |
| 69. A triangle has lattice points as vertices and contains no other lattice points. Prove that its area is 1/2. | |
| 70. Let S be the set of all positive multiples of k. Find an asymptotic expression for f(x) = ∑n∈S xn/n! [In other words, find g(x) in closed form so that f(x)/g(x) → 1 as n → ∞. What is the error term?] |
Donald J Newman, A problem seminar (Springer, problem books in mathematics, 1982). Highly recommended. It comes with short hints, giving the key idea and solutions which are concise but motivated (in other words, they often explain how you might have found the solution yourself).
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John Scholes
jscholes@kalva.demon.co.uk
25 Sep 1999