Chapter 5.2, Problem 111E

A general telescoping series is one in which all but the first few terms cancel out after summing a given number of successive terms.

111. Suppose that $a_n=c_{0}f\left(n\right)+f\left(n+1\right)+c_{2}f\left(n+2\right)+c_{3}f\left(n+3\right)+c_{4}f\left(n+4\right)$ . Where $f\left(n\right)\to 0$ as $n\to \infty$ . Find a condition on the coefficients $c_0,\mathrm{...},c_4$ that make this a general telescoping series.