ABSTRACT
Leonhard Euler (1707-1783) discovered his powerful “summation formula” in the early 1730s. He used it in 1736 to compute the first 20 decimal places for the alternating sum
(−1)g 2g + 1
= 1− 1 3 + 1
5 − 1 7 +− . . . = π
4 . (1.1)
Since, aside from the geometric series, very few infinite series then had a known sum, Euler’s remarkable sum enticed mathematicians like G. Leibniz (16461716) and the Bernoulli brothers Jakob (1654-1705) and Johann (1667-1748) to seek sums of other series, particularly the sum of the reciprocal squares. But it was L. Euler, within the next two decades up to 1750, who did a “broadening of the context” to formulate his “summation formula” for the general sum
F (g) = ∑
F (g) = ∑
F (g) (1.2)
(with n possibly infinite). More concretely, under the assumption of second order continuous derivatives of F on the interval [0, n], n ∈ N, Euler succeeded in finding the summation formula∑
F (g)− 1 2 (F (0) + F (n)) (1.3)
=
F (x) dx+ 1
12 (F ′(n)− F ′(0)) +
( −1 2 B2(x)
) ︸ ︷︷ ︸
F ′′(x) dx,
B2(x) = (x− ⌊x⌋)2 − (x− ⌊x⌋) + 1 6
(1.4)
is the “Bernoulli function” of degree 2. In particular, Euler’s new setting also encompassed the quest for closed formulas for sums of powers
kl ≃ ∫ n 0
xl dx, (1.5)
which had been sought since antiquity for area and volume investigations. In addition, this setting provided a canonical basis for the introduction of the Zeta function.
