Bài giải:

$\begin{align} & S=\frac{1}{5\times 9}+\frac{1}{9\times 13}+\frac{1}{13\times 17}+...+\frac{1}{41\times 45} \\ & =\frac{1}{4}\times \left( \frac{9-5}{5\times 9}+\frac{13-9}{9\times 13}+\frac{17-13}{13\times 17}+...+\frac{45-41}{41\times 45} \right) \\ & =\frac{1}{4}\times \left( \frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+\frac{1}{13}-\frac{1}{17}+...+\frac{1}{41}-\frac{1}{45} \right) \\  & =\frac{1}{4}\times \left( \frac{1}{5}-\frac{1}{45} \right) \\  & =\frac{2}{45} \\ \end{align}$