Useful Partial Sums

Question

The following formula: $$\sum_{k=m}^nf(k)=c(n-m)+\sum_{k=m}^\infty(f(k)-f(k+n))$$(Where $f\rightarrow c$) can be proven by telescoping the infinite sum in the RHS. The use of this formula is to expand the domain of $n$ in the LHS to all real numbers. So I was wondering whether there are important partial sums (i.e. they have applications to other areas, like in a non-trivial identity (no $+1-1$ or $\times1$ tricks) or a use in some Science) whose summand approaches a constant. This way, I could use the above formula to transform it into an infinite series. I could think of the Harmonic numbers and the Alternating Harmonic Numbers, but what else?