On the Keller limit and generalization

Abstract

Abstract Let c be any real number and let u n ( c ) = ( n + 1 ) ( 1 + 1 n + c ) n + c − n ( 1 + 1 n + c − 1 ) n + c − 1 − e . $$u_{n}(c)=(n+1) \biggl(1+\frac{1}{n+c} \biggr)^{n+c}-n \biggl(1+\frac {1}{n+c-1} \biggr)^{n+c-1}-e. $$ In this note, we establish an integral expression of u n ( c ) $u_{n}(c)$ , which provides a direct proof of Theorem 1 in (Mortici and Jang in Filomat 7:1535-1539, 2015).

Keywords

• Keller’s limit
• constant e
• integral expression