Series| Series |
Evaluating Series
> s.a. Feynman Diagrams.
* Method: If the integral
of the function \(f(z)\) along a rectangle in the complex plane vanishes
in the limit of a large rectangle, then we can calculate
∑k ∈ \(\mathbb Z\) f(k) = ∑poles of cot(πz) res(π cot(πz)) f(z) .
* Divergent series: A number of
methods have been tried, most notably Padé approximants, Borel-Padé
summation, Borel transformation with mapping, and order-dependent mapping.
@ Texts: Bromwich 47;
Knopp 51;
Adrian 06 [special series, I].
@ General references:
Varadarajan BAMS(07) [Euler's work];
Roy 11 [series and products from the XV to the XXI century;
r Isis(12)];
blog pt(14)feb [the sum of all positive integers equals −1/12];
Valean 19 [derivations and difficult cases];
Roy 21,
21
[history, and products].
@ Convergence acceleration: Amore JMAA(06)mp/04
[series for π, Catalan constant, Riemann zeta function, ...];
Caliceti et al PRP(07);
Bender & Heissenberg a1703-ln [and physics];
Costin & Dunne JPA(18)-a1705
[converting divergent series into rapidly convergent ones, and physics].
@ Divergent series: Parwani ht/00,
IJMPA(03)mp/02 [bounds];
Bellet mp/02 [finite results in perturbation series];
Zinn-Justin a1001-proc [order-dependent mapping];
Álvarez & Silverstone JPcomm(17)-a1705 [sum by educated match].
Convergence Criteria
* Comparison test:
∑n an
converges if there is a sequence {un} such that
∀n > N, an ≤
un and ∑n
un < ∞.
* Cauchy root test: ∑n
an converges if
(an)1/n
\(\le\) r < 1.
* D'Alembert or Cauchy ratio test:
∑n an
converges if an+1
/ an ≤ r < 1.
Taylor Series > s.a. analytic functions.
* For a function on R:
\[f(x) = \sum\nolimits_{n = 0}^\infty {1\over n!}\, {{\rm d}^n f\over{\rm d}x^n}\Big|_{x_0} (x-x_0)^n.\]
* For a function on a Lie group: If f : G → \(\mathbb C\), with Lie algebra \(\cal G\), expanding around h ∈ \(\cal G\),
f(hg) = ∑n = 0∞ (1/n!) (vn f)(h) ,
where g = exp γ, and v is the left-invariant
vector field generated by γ ∈ \(\cal G\).
* Examples:
\[ \def\ee{{\rm e}} \ee^x = \sum\nolimits_{k=0}^\infty {x^k\over k!} = \ee^a \sum\nolimits_{k=0}^\infty {1\over k!}\,(x-a)^k\;,\qquad \sinh x = \sum\nolimits_{k=0}^\infty {x^{2k+1}\over(2k+1)!}\;,\qquad \cosh x = \sum\nolimits_{k=0}^\infty {x^{2k}\over(2k)!}\]
\[ {1\over1+x} = 1 - x + x^2 - x^3 + x^4 + {\cal O}(x^5) \]
\[ \sqrt{1^{\vphantom1}+x} = 1+{\textstyle{1\over2}}\,x
- {\textstyle{1\over8}}\,x^2 + {\textstyle{1\over16}}\,x^3
- {\textstyle{5\over128}}\,x^4 + {\cal O}(x^5) \]
\[ \ln(1+x) = x - {\textstyle{1\over2}}\,x^2 + {\textstyle{1\over3}}\,x^3 - {\textstyle{1\over4}}\,x^4 + {\cal O}(x^5)\]
@ References: Sturzu mp/04 [for operator functions]; > s.a. numbers [continued fractions].
Other Types and Related Concepts > s.a. Asymptotic
Expansions; fourier analysis; functions;
sequences; summations.
* Geometric and related series:
∑n = 0∞ qn = (1−q)−1 , for |q| < 1 ; ∑n = 0∞ n qn = q/(1−q)2 .
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send feedback and suggestions to bombelli at olemiss.edu – modified 22 apr 2021