HMMT 2014 U26

For $1 \leq j \leq 2014$, define

$$b_{j}=j^{2014} \prod_{i=1, i \neq j}^{2014}\left(i^{2014}-j^{2014}\right)$$

where the product is over all $i \in\{1, \ldots, 2014\}$ except $i=j$. Evaluate

$$\frac{1}{b_{1}}+\frac{1}{b_{2}}+\cdots+\frac{1}{b_{2014}}$$