"Some sequences of Euler type, their convergences and their stability"

"The aim of this paper is to present some sequences of Euler type. We will explore the sequences $\left( F_{n}\right) _{n\geq 1},$ defined by $% F_{n}\left( x\right) =\sum_{k=1}^{n}f\left( k\right) -\int_{1}^{n+x}f\left( t\right) dt,$ for any $n\geq 1$ and $x\in \left[ 0,1\right] ,$ where $f$ is a local integrable and positive function defined on $\left[ 1,\infty \right) $. Starting from some particular example we will find that this sequence is uniformly convergent to a constant function. Also, we present a stability result."