Complete monotonicity and limit of a generalized Euler sequence

We prove that if $f:(0,\infty)\to\mathbb{R}$ is a completely monotonic function then the generalized Euler sequence $$(a_n)_{n\ge1},\quad a_n=f(1)+\cdots+f(n)-\int _1^n f(t)\,dt $$ is completely monotonic, and under appropriate conditions on f we obtain an explicit formula for its limit. Some particular cases of Stieltjes constants are studied. We also give representations for the sum of some generalized harmonic series.