A class of applications of AM-GM Inequality: From a 2004 Putnam competition problem to Lalescu’s sequence

Discussing a series of inequalities, B. Bollobás reminds us in [3] that Harald Bohr wrote: “All analysts spend half their time hunting through the literature for inequalities which they want to use but cannot prove. ” Fortunately, other inequalities can be reduced to techniques whose strategy of proof is familiar to us. This expository note has been inspired by the problem B2 from the 2004 edition of the W.L. Putnam competition. We will show that a natural context where this problem can be discussed is the area of applications of arithmetic mean-geometric mean (AM-GM) inequality. We conclude our note with a presentation of a classic problem, the Lalescu’s sequence. Using only elementary arguments, we can show that the sequence xn = 1 + 1n)n is increasing and convergent to a limit between 2 and 3 that we denote by e. The fact that xn is increasing is mentioned in [5], p. 37, application 35. In [7], this fact is proved as an application of the AM-GM inequality. To remind ourselves here of the proof that {xn}n∈N is increasing, we start with the AM-GM inequality. For a1, a2,..., an ≥ 0, we have: a1 + a2 +...+ an n ≥ (a1a2...an)1/n, with equality if and only if a1 = a2 =... = an. This inequality has many simple proofs. For example, a proof based on the concavity of the logarithmic function is presented in various sources, and the original reference is Jensen’s paper [6]. A proof based on induction, given by Cauchy in 1821, is presented in many sources, as for example in [3], pp. 1–2. We use the AM-GM inequality to show that {xn}n∈N is increasing. We apply this inequality to n + 1 positive real numbers, a1 = 1, a2 =... = an+1 = 1 + 1n. We get: 1 + n(1 + 1n) n+