Summary: The aim of this paper is to present some sequences of Euler type. We will explore the sequences \((F_n)_{n\ge 1}\), defined by \(F_n(x) =\sum^n_{k=1}f(k)-\int^{n+x}_1f (t)\, dt\), for any \(n\ge 1\) and \(x\in [0, 1]\), where \(f\) is a local integrable and positive function defined on \([1,\infty)\). Starting from some particular example we will find that this sequence is uniformly convergent to a constant function. Also, we present a stability result.