Abstract
In this chapter, we use the theory of summability of divergent series, presented earlier in Chap. 4, to derive the analogs of the Euler-Maclaurin summation formula for oscillating sums. These formulas will, in turn, be used to perform many remarkable deeds with ease. For instance, they can be used to derive analytic expressions for summable divergent series, obtain asymptotic expressions of oscillating series, and even accelerate the convergence of series by several orders of magnitude. Moreover, we will prove the notable fact that, as far as the foundational rules of summability calculus are concerned, summable divergent series behave exactly as if they were convergent.
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers.
Heinrich Hertz (1857–1894)
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Notes
- 1.
The summation formula of Theorem 5.1 is similar to, but is different from, Boole’s summation formula. Here, Boole’s summation formula states that \(\sum _{k=a}^{n} (-1)^k\,g(k) = \frac {1}{2} \sum _{k=0}^\infty \frac {E_k(0)}{k!} \big ((-1)^n f^{(k)}(n+1)+(-1)^a f^{(k)}(a)\big )\), where E k (x) are the Euler polynomials [BCM09, BBD89, Jor65].
- 2.
It is worth mentioning that many algorithms exist for accelerating the convergence of alternating series, some of which can sometimes yield several digits of accuracy per iteration. Interestingly, some of those algorithms, such as the one proposed in [CVZ00], were also found to be capable of “correctly” summing some divergent alternating series as well.
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Alabdulmohsin, I.M. (2018). Oscillating Finite Sums. In: Summability Calculus. Springer, Cham. https://doi.org/10.1007/978-3-319-74648-7_5
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