Fractions are a pain

Calculus Level 5

Find the value of \[1 - \frac{7}{12} + \frac{25}{288} - \frac{91}{5184} + \dots + (-1)^r \frac{3^r + 4^r}{r \times 12^r} + \dots \]

(Note: When \(r=0\), \((-1)^r \frac{3^r + 4^r}{r \times12^r}\) would be but for the sake of the problem, it's defined to be \(1\) here.)

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