46th Putnam 1985

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Problem A3

x is a real. Define ai 0 = x/2i, ai j+1 = ai j2 + 2 ai j. What is limn→∞ an n?

 

Solution

Answer: ex - 1.

an+1 n for x is the same as an n for x/2. So if we write an n as pn(x) then evidently pn is a polynomial and we have the relation pn+1(x) = 2pn(x/2) + pn(x/2)2 (*) and also p0(x) = x.

If we add 1 to both sides of (*), then it becomes 1 + pn+1(x) = (1 + pn(x/2) )2. Iterating: 1 + pn(x) = (1 + x/N)N, where N = 2n. But lim (1 + x/n)n = ex as n → ∞, so pn(x) → ex - 1.

 


 

46th Putnam 1985

© John Scholes
jscholes@kalva.demon.co.uk
7 Jan 2001