2015 CJC Promo Q4 (c)

A geometric series is given by

\(1+\frac{{{x}^{2}}-5}{4}+\frac{{{\left( {{x}^{2}}-5 \right)}^{2}}}{16}+...+\frac{{{\left( {{x}^{2}}-5 \right)}^{n}}}{{{4}^{n}}}+...\)

1. Find the range of values of \(x\) such that the series converges.[2]
2. If the ratio of the sum of the first five even-numbered terms to the sum of the first five odd-numbered terms is \(\frac{x}{3}\), determine the sum to infinity of the geometric series.[3]