(1)Problem 11748 (American Mathematical Monthly, Vol.121, January 2014) Proposed by C

Problem 11748

(American Mathematical Monthly, Vol.121, January 2014)

Proposed by C. Lupu (USA) and T. Lupu (Romania).

Is there a sequencea₁, a₂, . . . of positive real numbers such that P^∞

n=11/anconverges, and Qn

k=1ak< nⁿ for alln?

Solution proposed by Roberto Tauraso, Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy.

No, such sequence does not exist, otherwise, by Carleman’s inequality, we would have the following contradiction

+∞ = X∞ n=1

1 n≤

X∞ n=1

1 (Qn

k=1ak)^1/n

≤e X∞ n=1

1 an

< +∞