In mathematics, an infinite geometric series of the form
\( \sum _{{k=0}}^{\infty }ar^{k}=a+ar+ar^{2}+ar^{3}+\cdots \)
is divergent if and only if | r | ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case
\( \sum _{{k=0}}^{\infty }ar^{k}={\frac {a}{1-r}}. \)
This is true of any summation method that possesses the properties of regularity, linearity, and stability.
Examples
In increasing order of difficulty to sum:
1 − 1 + 1 − 1 + · · ·, whose common ratio is −1
1 − 2 + 4 − 8 + · · ·, whose common ratio is −2
1 + 2 + 4 + 8 + · · ·, whose common ratio is 2
1 + 1 + 1 + 1 + · · ·, whose common ratio is 1.
Motivation for study
It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method sums Σzn to 1/(1 - z) for all z in a subset S of the complex plane, given certain restrictions on S, then the method also gives the analytic continuation of any other function f(z) = Σanzn on the intersection of S with the Mittag-Leffler star for f.[1]
Summability by region
Open unit disk
Ordinary summation succeeds only for common ratios |z| < 1.
Closed unit disk
Cesàro summation
Abel summation
Larger disks
Euler summation
Half-plane
The series is Borel summable for every z with real part < 1. Any such series is also summable by the generalized Euler method (E, a) for appropriate a.
Shadowed plane
Certain moment constant methods besides Borel summation can sum the geometric series on the entire Mittag-Leffler star of the function 1/(1 − z), that is, for all z except the ray z ≥ 1.[2]
Everywhere
Notes
Korevaar p.288
Moroz p.21
References
Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X.
Moroz, Alexander (1991). "Quantum Field Theory as a Problem of Resummation". arXiv:hep-th/9206074.
Sequences and series
Integer
sequences
Basic
Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced (list)
Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number
Fibonacci spiral with square sizes up to 34.
Properties of sequences
Cauchy sequence Monotone sequence Periodic sequence
Properties of series
Convergent series Divergent series Conditional convergence Absolute convergence Uniform convergence Alternating series Telescoping series
Explicit series
Convergent
1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2s+ 1/3s + ... (Riemann zeta function)
Divergent
1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) Infinite arithmetic series 1 − 2 + 3 − 4 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series
Taylor series Power series Formal power series Laurent series Puiseux series Dirichlet series Trigonometric series Fourier series Generating series
Hypergeometric
series
Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric series
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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