Polynomials and infinite sum

Algebra Level 2

The polynomial \(p(x) = x^2 - ax + \frac14\), with \(a\) a positive real, has two real roots (not necessarily distinct) \(r_1\) and \(r_2\) such that \(\left| r_1 \right| \le \frac12\) and \(\left| r_2 \right| \le \frac12\). Find the value of: \[r_1 + r_2 + {r_1}^2 + {r_2}^2 + {r_1}^3 + {r_2}^3 + \dots = \sum_{k=1}^\infty \left({r_1}^k + {r_2}^k\right)\]

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