It seems that under the light of Ramanujan Summation the following is plausible:
$$1 + {2^{2n - 1}} + {3^{2n - 1}} + \cdots = - \frac{{{B_{2n}}}}{{2n}}(\Re)$$
Alas, I can't really find any concrete definition of Ramanujan Summation. Could someone provide a small explanation?