For alternating series, $|S_N-S||a_{N+1}|.$ What

$is$ the smallest value $ofN$ that approximates the

series

$S={\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n+1}}{(n+78)(n+75)}$

$to$ within $an$ error $ofat$ most ${10}^{-4}?$

$N=$

$S$~~