Laplace II: Steps, Impulses & Convolution · Superchallenge problem

A damped oscillator at rest is kicked with a unit impulse every $\pi$ seconds: $y''+2y'+2y=\displaystyle\sum_{k=0}^{\infty}\delta(t-k\pi)$, $y(0^-)=y'(0^-)=0$. Find $\displaystyle\lim_{n\to\infty}y\!\left(n\pi+\dfrac{\pi}{2}\right)$.