Problem 11773
(American Mathematical Monthly, Vol.121, April 2014) Proposed by M. Omarjee (France).
Given a positive real number a0, let an+1= exp (−Pn
k=0ak) for n ≥ 0. For which values of b does P∞
n=0(an)b converge?
Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.
We note that
an+1= exp −
n−1
X
k=0
ak
!
e−an= ane−an= f (an)
where f (x) = xe−x. Since, f (0) = 0 and 0 < f (x) < x for x > 0, it follows that the sequence {an}n≥0is positive and strictly decreasing to 0. Moreover
f (x) = x − x2+ x3/2 + o(x3)
and by Theorem 2 in Effective asymptotic for some nonlinear recurrences and almost doubly- exponential sequences, by Ionascu and Stanica, Acta Mathematica Universitatis Comenianae (2004), we have that
an = 1
n−ln(n)
2n2 + o ln(n) n2
. Thus,P∞
n=0(an)bconverge iffP∞
n=11/nb converge, that is for b > 1.
