TY - CHAP
T1 - Oscillating Finite Sums
AU - Alabdulmohsin, Ibrahim
N1 - KAUST Repository Item: Exported on 2020-10-01
PY - 2018/3/8
Y1 - 2018/3/8
N2 - 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.
AB - 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.
UR - http://hdl.handle.net/10754/627461
UR - https://link.springer.com/chapter/10.1007%2F978-3-319-74648-7_5
U2 - 10.1007/978-3-319-74648-7_5
DO - 10.1007/978-3-319-74648-7_5
M3 - Chapter
SN - 9783319746470
SP - 93
EP - 113
BT - Summability Calculus
PB - Springer Nature
ER -