We perform Lagrange interpolation on the polynomial $P(x)=1$ through the points $1^{2014}, 2^{2014}, \ldots, 2014^{2014}$. We have $1=P(x)=\sum_{j=1}^{2014} \frac{\prod_{i=1, i \neq j}^{2014}(x-i^{2014})}{\prod_{i=1, i \neq j}^{2014}(j^{2014}-i^{2014})}$. Thus, $1=P(0)=\sum_{j=1}^{2014} \frac{((-1)^{2013}) \frac{2014!^{2014}}{j^{2014}}}{(-1)^{2013} \prod_{i=1, i \neq j}^{2014}(i^{2014}-j^{2014})}$ which equals $2014!^{2014} \sum_{j=1}^{2014} \frac{1}{j^{2014} \prod_{i=1, i \neq j}^{2014}(i^{2014}-j^{2014})}=2014!^{2014}\left(\frac{1}{b_{1}}+\frac{1}{b_{2}}+\cdots+\frac{1}{b_{2014}}\right)$ so the desired sum is $\frac{1}{2014!^{2014}}$.