Solve what? You just gave the answer. No, it not, strictly speaking a "harmonic series"- that is specifically 1+ 1/2+ 1/3+ 1/4+...

But you are correct that this series converges since n+2> n and n+3> n

1/(n(n+2)(n+3))< 1/n^3 and, by the "p-text" you cite, 1/n^3 converges so 1/(n(n+2)(n+3)) converges by the "comparison test".