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