Series Convergence

Calculus Level 5

\[ \large \sum_{n=1}^\infty \dfrac{(a-3\cos^2 n)\sin^2 n}{\sqrt n} \]

Let \(a\) be the constant value for which the series above converges.

If \(a\) can be expressed as \( \dfrac bc\), where \(b\) and \(c\) are integers, with \(b\) positive, find \(b+c\).

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