English: A smoothed version of the divergent series 1 + 2 + 3 + 4 + ⋯. This diagram is analogous to Sum1234Plain.svg, but a smoothing function

${\displaystyle \eta (x)=\exp \left(1-{\frac {1}{1-x^{2}}}\right)}$ has been applied, so that the blue curve is the graph of the function ${\displaystyle \sum _{n=1}^{\infty }n\eta (n/x)}$. The horizontal scale has also been compressed so that the asymptotic growth of the function is x²/2, to aid comparison with the original series.

The blue curve asymptotically approaches the parabola ${\displaystyle -{\frac {1}{12}}+{\frac {x^{2}}{2}}}$, which is depicted in the companion diagram Sum1234Asymptote.svg.