Let \(a_n=\sin{\left(\dfrac{\pi}2-n\pi\right)}+\cos{n\pi}\) and \(b_n=10\cos{\left(2n\pi-\dfrac{\pi}3\right)}\). Find the value of \[\large\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{a_n}{b_n}\right)^n\]

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