The author proves the following theorem: If \(f\) and \(f^{(r)}\) are of bounded variation on \([n,\infty)\), and \(\lim f^{(\nu)}(x)= 0\) for \(\nu\in [0,r]\) as \(x\to +\infty\), then for \(x\in [-\pi,\pi]\setminus \{0\}\) the following relation is valid: \[ \begin{multlined} \sum^\infty_{k= n} f(k)e^{ikx}= \int^\infty_n f(u) e^{iux}du+ {1\over 2} f(n) e^{inx}+\\ e^{inx} \sum^{r- 1}_{p= 0} {(-1)^{p+ 1}\over p!} h^{(p)}(x) f^{(p)}(n)+ {Q\over \pi^r} V^\infty_n(f^{(r)}),\end{multlined} \] where \(h(x)= {1\over x}-{1\over 2}\text{ cot }{x\over 2}\), \(| Q|\leq 3\) and \(V^\infty_n(f^{(r)})\) is the total variation of \(f^{(r)}\).