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[Texts]## 40: Sequences, series, summability |

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.

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.

- 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.

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

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

- 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) ).
- 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)?
- 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.
- 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)
- 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").
- Obtaining closed forms for the sum sum( 1/n^2 ) = pi/6 and similar sums

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org