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]; blog pt(14)feb [the sum of all positive integers equals −1/12].
@ 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 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 et al a1705 [sum by educated match].

Convergence Criteria
* Comparison test: ∑n an converges if there is a sequence {un} such that ∀n > N, anun 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/anr < 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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