Induction · Superchallenge problem

The claim $\displaystyle \sum_{i=1}^{n} \frac{1}{i^3} < \frac32$ cannot be proved by a direct induction (the step would need the sum up to $k$ to be well below $\tfrac32$). Instead prove $\displaystyle \sum_{i=1}^n \frac{1}{i^3} \le c - \frac{1}{2n^2}$. The step works for every $c$; the base $n = 1$ decides it. What is the smallest $c$ that works?