This is similar to Borel's integral summation method, except that the Borel transform need not converge for all t, but
converges to an analytic function of t near 0 that can be analytically continued along the positive real axis.
The Borel sum of 1- 2 + 4- 8 + ⋯ is also 1/3;
when Émile Borel introduced the limit formulation of Borel summation in 1896,
this was one of his first examples after 1- 1 + 1- 1 + ⋯ Leibniz pp. 205-207 Knobloch pp.