Not to detract from your point, but you don't need the expansion for that. Directly substituting 𝟏 in the definitional equation works just fine:

    L a = 𝟏 + a × (L a)
    L 𝟏 = 𝟏 + 𝟏 × (L 𝟏)
    L 𝟏 = 𝟏 +     (L 𝟏)
    L 𝟏 = 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟏 + ...
    ℕ   = 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟏 + 𝟏 + ...

The final expansion is worth writing, however, since it notes how there are unboundedly many naturals, each a different component in this infinite disjoint sum.