Write down and simplify the first three terms of the Maclaurin series for \(\mathrm { e } ^ { 2 x }\).
Hence show that the Maclaurin series for
$$\ln \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)$$
begins \(\ln a + b x ^ { 2 }\), where \(a\) and \(b\) are constants to be found.