On Taylor coefficients of analytic function related with Euler number

Abstract

We consider a classical construction of second remarkable limit. We pose a question on asymptotically sharp description of the character of such approximation of the number $e$. In view of this we need the information on behavior of the coefficients in the power expansion for the function $f(x)=e^{-1}\,(1+x)^{1/x}$ converging in the interval $-1-1$ having the same Taylor coefficients and being analytic in the complex plane $\mathbb{C}$ with the cut along $(-\infty,\,-1]$. By the methods of the complex analysis we obtain an integral representation for $a_n$ for each value of the parameter $n\in\mathbb{N}$. We prove that $a_n\rightarrow 1/e$ as $n\rightarrow\infty$ and find the convergence rate of the difference $a_n-1/e$ to zero. We also discuss the issue on choosing the contour in the integral Cauchy formula for calculating the Taylor coefficients $(-1)^n\,a_n$ of the function $f(z)$. We find the exact values of arising in calculations special improper integrals. The results of the made study allows us to give a series of general two-sided estimates for the deviation $e-(1+x)^{1/x}$ consistent with the asymptotics of $f(x)$ as $x\to 0$. We discuss the possibilities of applying the obtained statements.