2021 HCI Promo Q6

The sum of the first \(n\) terms of a series is given by the expression \({{\mathrm{e}}^{2}}-{{\left( -2 \right)}^{n}}\left( {{\mathrm{e}}^{2-n}} \right)\).

1. State the first term of the series in terms of \(\mathrm{e}\).[1]
2. By finding the \({{n}^{\mathrm{th}}}\) term of the series, show that this is a geometric series.[3]
3. Explain why the sum to infinity, \(S\), of the series exists, and determine the exact value of \(S\).[3]