Nonhomogeneous Equations & Resonance · Superchallenge problem

For each positive integer $n$, let $y_n$ solve $y''+y=\cos\!\left(\left(1+\dfrac{1}{2n}\right)t\right)$ with $y(0)=0$, $y'(0)=0$. Compute $\displaystyle\lim_{n\to\infty}\dfrac{y_n(2n\pi)}{n}$.