Universal method for summation of series, further developed
In this paper, Euler examines the formal series f(a) – f(a+h) + f(a+2h) – f(a+3h) + f(a+4h) Â± .... Apparently, he looks at this series in E352 and E617 as well. Here, Euler uses a power series f(0) + f(1)x + f(2)x2 + f(3)x3 + ..., which is obviously what he's looking for in the case a=0, h=1 and x=-1. He does Taylor series expansions at every value, 0, 1, 2,... , sums them all, and combines terms and gets an answer reminiscent of the one in E352.
Original Source Citation
Commentarii academiae scientiarum Petropolitanae, Volume 8, pp. 147-158.
Opera Omnia Citation
Series 1, Volume 14, pp.124-137.