Inequalities for residuals of power expansions for the exponential function and completely monotone functions
Let \(I_n(x)= \sum^{+\infty}_{k=n+1} (-1)^kx^k/k!\) \((x>0)\), \(n=1,2,\dots\) . Put \[ J_n(x):= I_{n-1}(x) I_{n+1}(x)/I^2_n(x). \] The author proves that: \[ n/(n+1)\leq J_n(x)\leq (n+1)/(n+ 2)\tag{1} \] for all \(x>0\), \(n\geq 1\); and that \(\lim_{x\to 0} J_n(x)= (n+1)/(n+2)\), \(\lim_{x\to\infty} J_n(x)= n/(n+ 1)\); \[ |I_n(x)|= x^{n+1}/(n+ 1)!(1+ x/(n+ a(x))),\tag{2} \] where \(a(x)\in (1,2)\) \((x>0)\) and \(\lim_{x\to 0} a(x)= 2\), \(\lim_{x\to\infty} a(x)= 1\); (3) The left-side inequality of (1) is valid (in some sense) for a general completely monotonic function \(f:(0,\infty)\to\mathbb{R}\). The proof is based essentially on Maclaurin's expansion with integral remainder term, as well as certain properties of the confluent hypergeometric function.
inequalities for the exponential function
scientific article; zbMATH DE number 1053555