By Ramanujan,
$$1+2+3+4+ \dots = -\frac{1}{12}$$
Observe, however
$$y = \sum_{i=0}^\infty 13^i = 1 + \sum_{i=1}^\infty 13^i = 1 + 13\sum_{i=0}^\infty 13^i$$
This implies
$$y = 1 + 13y \longrightarrow y = -\frac{1}{12}$$
Therefore,
$$1+2+3+4+5+ \dots = 1+13+13^2 + 13^3 + 13^4 + \dots$$
Q.E.D.