English: A smoothed version of the divergent series 1 + 1 + 1 + 1 + ⋯. This diagram is analogous to Sum1111Plain.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 ${\displaystyle \sum _{n=1}^{\infty }\eta (n/x)}$. The horizontal scale has also been compressed so that the asymptotic growth of the function is x, to aid comparison with the original series.

The blue curve asymptotically approaches the line ${\displaystyle -{\frac {1}{2}}+x}$, which is depicted in the companion diagram Sum1111Asymptote.svg.