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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 40: Sequences, series, summability

Introduction

Sequences and series are really just the most common examples of limiting processes; convergence criteria and rates of convergence are as important as finding "the answer". (In the case of sequences of functions, it's also important do find "the question"!) Particular series of interest (e.g. Taylor series of known functions) are of interest, as well as general methods for computing sums rapidly, or formally. Series can be estimated with integrals, their stability can be investigated with analysis. Manipulations of series (e.g. multiplying or inverting) are also of importance.

History

Applications and related fields

Sequences are discussed here, but for sequences of integers and their number-theoretic properties, see number theory.

Finite trigonometric sums are treated in 11L: Exponential sums and character sums.

The general question of whether or not a function defined by a series can be evaluated simply in terms of "known" functions is delicate; by analogy with differential equations, it is possible to deduce some answers using the tools of 12H: Differential and difference algebra

Subfields

• 40A: Convergence and divergence of infinite limiting processes
• 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)
• 40C: General summability methods
• 40D: Direct theorems on summability
• 40E: Inversion theorems
• 40F05: Absolute and strong summability
• 40G: Special methods of summability
• 40H05: Functional analytic methods in summability
• 40J05: Summability in abstract structures, See also 43A55, 46A35, 46B15

This is one of the smallest areas in the Math Reviews database.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Software and tables

• Symbolic Summation package for Mathematica
• Sloane's sequence server (identifies an integer sequence from the first few terms).

Other web sites with this focus

• Here are the AMS and Goettingen resource pages for area 40.

Selected topics at this site

• Raabe's test for convergence of a series
• Sensitive tests for (conditional) convergence of a real series
• Using the saddle point method to estimate an alternating sum.
• The darnedest series arise in applied problems. Here was a request to sum: Sum[ v^(i-1)*Exp(-Lv)*((v-1)^(j-i)*L^j)/j!(i+1) ,1 \le i \le j < \infty]
• The Euler-Maclaurin formula, and other suggestions for computing partial sums of Sum( f(n) ).
• Euler-Maclaurin-summation technique
• Poisson's summation formula
• A citation on speeding up convergence of series.
• Speeding the convergence of a slowly converging series (via integral test).
• Example: speeding up the convergence of sum 1/(n * (ln(n))^2 ) .
• The Levin transform (for speeding up convergence of infinite sums), with code fragment.
• Computing terms of the Laurent series of 1/(1-x*cot(x))
• A closed for is sought for a sequence defined recursively by x_{n+1}=x_n-(x_n^2)/n
• A recursively-defined sequence akin to the Bernoulli numbers: a_k = 1 - 2\sum_{j=0}^{k-1} {k\choose j} a_j
• Are there methods for symbolic summation (as for symbolic integration)?
• Putting a recursive sequence into closed form (with and without Maple)
• Citations for methods of summation (Moenck, Zeilberger, Koepf, Karr, etc.)
• A "Theory of symbolic summation" (the book "A = B").
• Algorithms for evaluating some types of power series in closed form
• Evaluating a power series in closed form using Zeilberger's algorithm (EKHAD package)
• Formal characterization of "closed-form solutions" to differential equations, integrals, and summation problems
• Axiomatic treatment of infinite series
• Deriving closed-form expressions for infinite sums (experimentally) with PSLQ
• Are there any methods for finding closed formulas for 2-dimensional recurrence problems in general?
• An example of a recurrence relation defining a sequence growing doubly exponentially: f(n)=f(n-1)+f(n-1)f(n-2)
• An example of a series expansion with very delicate convergence.
• Levy-Steinitz theorem: conditionally convergent sums of vectors can be rearranged to sum to many values
• Unit vectors which sum to zero can be reordered to keep partial sums small
• Evaluating an infinite sum from probability -- Sum( a^(d-N) (1-a)^N d! / (N-1)!(d-N)!, d > N )
• Generalities on "finding the next term in this sequence" problems.
• Pointer to the excellent sequence server ("what sequence begins as follows...?")
• Typical example (from physics) of estimating rate of growth of a series.
• Generalities on convergence of series of matrices (and diagonalization)
• Taylor series convergence at endpoints, using Abel summation as Stieltjes integrals
• Application of Lagrange Inversion Formula for power series
• Obtaining closed forms for the sum sum( 1/n^2 ) = pi/6 and similar sums