Summation of Grandi's Series - Abel Sum

Abel Sum

Abel summation is similar to Euler's attempted definition of sums of divergent series, but it avoids Callet's and N. Bernoulli's objections by precisely constructing the function to use. In fact, Euler likely meant to limit his definition to power series, and in practice he used it almost exclusively in a form now known as Abel's method.

Given a series a₀ + a₁ + a₂ + · · ·, one forms a new series a₀ + a₁x + a₂x2 + · · ·. If the latter series converges for 0 < x < 1 to a function with a limit as x tends to 1, then this limit is called the Abel sum of the original series, after Abel's theorem which guarantees that the procedure is consistent with ordinary summation. For Grandi's series one has