Conditional convergence

Conditional convergenceMain

Community hub

7 pages, 0 posts

0 subscribers

Recent from talks

Be the first to start a discussion here.

Recent from talks

Be the first to start a discussion here.

Contribute something

About hubMembersContent overviewUpdatesRules

Main reference articles

Conditional convergence

View on Wikipedia

from Wikipedia

In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.

Definition

[edit]

More precisely, a series of real numbers ${\textstyle \sum _{n=0}^{\infty }a_{n}}$ is said to converge conditionally if ${\textstyle \lim _{m\rightarrow \infty }\,\sum _{n=0}^{m}a_{n}}$ exists (as a finite real number, i.e. not $\infty$ or $-\infty$ ), but ${\textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=\infty .}$

A classic example is the alternating harmonic series given by $1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\sum \limits _{n=1}^{\infty }{(-1)^{n+1} \over n},$ which converges to $\ln(2)$ , but is not absolutely convergent (see Harmonic series).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem. Agnew's theorem describes rearrangements that preserve convergence for all convergent series.

The Lévy–Steinitz theorem identifies the set of values to which a series of terms in Rⁿ can converge.

Indefinite integrals may also be conditionally convergent. A typical example of a conditionally convergent integral is (see Fresnel integral) $\int _{0}^{\infty }\sin(x^{2})dx,$ where the integrand oscillates between positive and negative values indefinitely, but enclosing smaller areas each time.

References

[edit]

Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).

Sequences and series

List of mathematical 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 sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Properties of sequences

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

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 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric series

Category

Revisions and contributors Edit on Wikipedia Read on Wikipedia

View on Grokipedia

from Grokipedia

In mathematics, conditional convergence refers to the convergence of an infinite series

\sum a_n

where the series converges, but the corresponding series of absolute values

\sum |a_n|

diverges.^[1]^[2] This contrasts with absolute convergence, where

\sum |a_n|

converges, implying the original series converges regardless of term order or sign changes.^[3]^[4] A classic example is the alternating harmonic series

\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}

, which converges to

\ln 2

by the alternating series test, but its absolute counterpart

\sum_{n=1}^\infty \frac{1}{n}

diverges as the harmonic series.^[5]^[6] Conditionally convergent series exhibit delicate behavior, as their convergence relies on cancellations between positive and negative terms, making them sensitive to rearrangements.^[7] The Riemann rearrangement theorem, proved by Bernhard Riemann in 1852, highlights this fragility: for any conditionally convergent series and any real number

L

, there exists a rearrangement of its terms that converges to

L

, or even to

\pm \infty

.^[8]^[9] This theorem underscores the importance of distinguishing conditional from absolute convergence in analysis, as absolute convergence preserves the sum under permutations, while conditional does not.^[3]

Convergence Basics

Absolute Convergence

In mathematics, a series

\sum_{n=1}^\infty a_n

of real or complex numbers is said to converge absolutely if the series

\sum_{n=1}^\infty |a_n|

of the absolute values converges to a finite limit.^[10] This condition ensures a stronger form of convergence than ordinary (or conditional) convergence, where the partial sums approach a limit without regard to the signs of the terms. Absolute convergence implies ordinary convergence because, for m > n, |s_m - s_n| = |∑{k=n+1}^m a_k| ≤ ∑{k=n+1}^m |a_k|, and the right-hand side approaches 0 as n, m → ∞ since ∑ |a_n| converges; thus, {s_n} is a Cauchy sequence and converges to some finite S with |S| ≤ ∑_{n=1}^∞ |a_n|. A key property is that absolute convergence is invariant under permutations of the terms, meaning the sum remains the same regardless of the order in which the terms are added.^[10] The concept of absolute convergence was first rigorously defined by Augustin-Louis Cauchy in his 1821 work Cours d'analyse, where he distinguished it from ordinary convergence to address issues in series summation.^[11] Karl Weierstrass later emphasized its importance in the mid-19th century through his lectures on function theory, highlighting its role in ensuring robust behavior of series under rearrangements and in uniform convergence tests.^[12] A classic example is the geometric series

\sum_{n=0}^\infty r^n

for

|r| < 1

, which converges absolutely because

\sum_{n=0}^\infty |r|^n = \sum_{n=0}^\infty |r|^n

is also a geometric series with ratio

|r| < 1

, summing to

\frac{1}{1 - |r|}

.^[13] In contrast, conditional convergence occurs when

\sum a_n

converges but

\sum |a_n|

diverges, making the sum sensitive to term order.^[10]

Conditional Convergence

In mathematics, particularly in the study of infinite series, conditional convergence describes a scenario where a series converges, but only in a manner dependent on the specific order of its terms. A series

\sum_{n=1}^\infty a_n

is said to converge conditionally if the sequence of its partial sums

s_n = \sum_{k=1}^n a_k

converges to a finite limit

S

, yet the series does not converge absolutely.^[14] This contrasts with absolute convergence, where the series

\sum_{n=1}^\infty |a_n|

converges, providing a stronger form of convergence that implies ordinary convergence regardless of term arrangement.^[15] Formally,

\sum_{n=1}^\infty a_n

converges conditionally to

S

\lim_{n \to \infty} s_n = S

and

\lim_{n \to \infty} \sum_{k=1}^n |a_k| = \infty

.^[14] Here, the failure of absolute convergence serves as the defining prerequisite, as established in standard real analysis texts: every absolutely convergent series is convergent, making conditional convergence the residual case of convergence without this absolute property.^[15] A key implication of conditional convergence is its sensitivity to the order of terms; unlike absolutely convergent series, rearrangements of the terms in a conditionally convergent series can yield a different sum or even cause divergence.^[14] This order dependence arises precisely because the absolute series diverges, allowing the partial sums to be influenced by how positive and negative terms are interleaved.^[15] The distinction from absolute convergence underpins the basic characterization of conditional convergence through a simple proof sketch: since absolute convergence implies convergence via the triangle inequality—specifically, for

m > n

|s_m - s_n| \leq \sum_{k=n+1}^m |a_k|

, which approaches 0 as

n, m \to \infty

\sum |a_k|

converges—any convergent series that lacks absolute convergence must be conditionally convergent.^[16]

Examples and Illustrations

Alternating Harmonic Series

The alternating harmonic series is defined as the infinite series

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots.

This series serves as the canonical example of conditional convergence in the context of infinite series.^[17] The convergence of the alternating harmonic series follows from the alternating series test, which states that an alternating series

\sum (-1)^{n+1} b_n

with

b_n > 0

converges if the sequence

\{b_n\}

is decreasing and

\lim_{n \to \infty} b_n = 0

. Here,

b_n = 1/n

, which is positive, decreasing, and approaches 0 as

n \to \infty

. The proof relies on showing that the sequence of partial sums is bounded and monotonic in subsequences: the odd-indexed partial sums decrease and are bounded below by 0, while the even-indexed partial sums increase and are bounded above by 1, implying both converge to the same limit by the monotone convergence theorem.^[18] The sum of the series equals

\ln 2 \approx 0.693147

. This result arises as the special case

x=1

of the Mercator series (Taylor expansion of

\ln(1+x)

)

\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} = \ln(1+x)

for

|x| < 1

, extended to

x=1

by Abel summation since the series converges at the endpoint of its interval of convergence

[-1, 1]

.^[17]^[19] The series does not converge absolutely because the absolute value series

\sum_{n=1}^{\infty} \frac{1}{n}

is the harmonic series, which diverges. The divergence of the harmonic series can be established using the integral test: consider

f(x) = 1/x

, which is positive, continuous, and decreasing on

[1, \infty)

; the improper integral

\int_1^{\infty} \frac{1}{x} \, dx = \lim_{b \to \infty} \ln b = \infty

diverges, so the series diverges by comparison, as the partial sums exceed the integral from 1 to

N+1

. Alternatively, it is a

p

-series with

p=1 \leq 1

, which diverges.^[20]^[21] For approximation, the error when truncating at the

n

th partial sum

S_n = \sum_{k=1}^n \frac{(-1)^{k+1}}{k}

is bounded by the magnitude of the next term:

|S - S_n| \leq \frac{1}{n+1}

, with the true sum

S = \ln 2

lying between

S_n

and

S_n + (-1)^{n+1}/(n+1)

. An explicit formula for the partial sum is

S_n = \ln 2 + (-1)^{n+1} \int_0^1 \frac{x^n}{1+x} \, dx,

where the integral term represents the remainder and alternates in sign while decreasing to 0.^[18]^[22] Numerical illustration of the partial sums approaching

\ln 2 \approx 0.693147

is shown below for the first six terms:

$n$	Partial Sum $S_n$	Approximation Error
1	1.000000	0.306853
2	0.500000	0.193147
3	0.833333	0.140186
4	0.583333	0.109814
5	0.783333	0.090186
6	0.616667	0.076480

The partial sums oscillate around

\ln 2

, with even sums below and odd sums above, and the amplitude of oscillation diminishes.^[17]

Other Standard Examples

Another standard example of a conditionally convergent series is the alternating logarithmic series

\sum_{n=2}^\infty \frac{(-1)^{n+1}}{n \ln n}

. This series converges by the alternating series test, as the terms

\frac{1}{n \ln n}

are positive, decreasing, and approach zero as

n \to \infty

. However, the absolute series

\sum_{n=2}^\infty \frac{1}{n \ln n}

diverges, which follows from the integral test applied to

\int_2^\infty \frac{dx}{x \ln x} = \ln(\ln x) \big|_2^\infty = \infty

∫2∞xlnxdx=ln(lnx)

2∞=∞.^[23] A variant of the p-series provides further examples of conditional convergence. For $0 < p \leq 1$ , the alternating p-series $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^p}$ converges by the alternating series test, since the terms $\frac{1}{n^p}$ decrease to zero. The absolute series $\sum_{n=1}^\infty \frac{1}{n^p}$ diverges as a p-series with $p \leq 1$ . Thus, the series converges conditionally in this range.^[24] In the context of Fourier series, conditional convergence appears in expansions of functions with discontinuities, such as the sawtooth wave defined by $f(x) = x$ for $-\pi < x < \pi$ (extended periodically). Its Fourier series is $\sum_{n=1}^\infty \frac{(-1)^{n+1} 2 \sin(nx)}{n}$ , which converges pointwise to $f(x)$ at continuity points but not absolutely, because the absolute series $\sum_{n=1}^\infty \frac{2}{n}$ diverges like the harmonic series. At discontinuities, the partial sums converge to the average of the left and right limits.^[25] Conditionally convergent series can be constructed generally by separating terms into positive and negative parts that each diverge, yet arranging them so the overall partial sums remain bounded and converge. For instance, start with divergent series of positive terms $\sum a_k = \infty$ with $a_k > 0$ and negative terms $\sum b_m = -\infty$ with $b_m < 0$ , then interleave them to control the partial sums, ensuring convergence while the absolute sum diverges. This approach underlies the flexibility shown in rearrangement theorems.^[7]

Theoretical Properties

Riemann Rearrangement Theorem

The Riemann rearrangement theorem, also known as the Riemann series theorem, was discovered by the German mathematician Bernhard Riemann during his work on the convergence of trigonometric series in the early 1850s and published posthumously in 1867 as part of his habilitation thesis.^[26] This result underscores a fundamental difference between absolute and conditional convergence, revealing that the order of terms in a conditionally convergent series profoundly affects its sum, a pathology absent in absolutely convergent series.^[27] The theorem states that if

\sum_{n=1}^\infty [a_n](/page/List_of_math_rock_groups)

is a conditionally convergent series of real numbers, then for every real number

r \in \mathbb{R}

and every

\varepsilon > 0

, there exists a rearrangement

\sum_{n=1}^\infty a_{\pi(n)}

(where

\pi

is a permutation of the natural numbers) such that the partial sums

s_m = \sum_{n=1}^m a_{\pi(n)}

satisfy

|s_m - r| < \varepsilon

for all sufficiently large

m

.^[26] To outline the proof, first separate the terms into positive and negative parts: let

P = \{a_n > 0\}

with

\sum_{a_n \in P} a_n = +\infty

and

N = \{a_n < 0\}

with

\sum_{a_n \in N} a_n = -\infty

, which follows from the conditional convergence assumption.^[27] Construct the rearrangement by starting from the zero partial sum and alternately adding blocks of positive terms until the sum exceeds

r

by less than

\varepsilon

, then adding blocks of negative terms until the sum falls below

r

by less than

\varepsilon

, repeating this process indefinitely. Since the terms

a_n \to 0

, the overshoots and undershoots remain bounded by

\varepsilon

, ensuring the partial sums converge to

r

.^[27] A key corollary is that rearrangements of a conditionally convergent series can also be made to diverge to

+\infty

-\infty

, or to oscillate without converging, by similar block constructions that force the partial sums to unbounded regions or bounded but non-convergent sets.^[27]

Uniqueness of Sums Under Rearrangement

In the case of absolute convergence, the sum of an infinite series

\sum a_n

remains unchanged under any rearrangement of its terms. This invariance holds because the absolute convergence ensures that the partial sums of the rearranged series remain close to the original sum, as the tails of the series of absolute values can be made arbitrarily small uniformly across rearrangements.^[28] By contrast, conditionally convergent series exhibit non-uniqueness of sums under arbitrary rearrangements, as demonstrated by the Riemann rearrangement theorem, which allows rearrangements to achieve any real value or even divergence. However, uniqueness can be restored under more restricted classes of permutations. For instance, rearrangements with bounded displacement—where the permutation

\sigma

satisfies

|\sigma(n) - n| \leq K

for some fixed integer

K

and all

n

—preserve the original sum even for conditionally convergent series, since such permutations limit the deviation in partial sums to a controlled amount.^[29] A special case arises with series having non-negative terms. If such a series

\sum a_n

with

a_n \geq 0

converges, it does so absolutely, as the absolute values coincide with the terms themselves, ensuring the sum is invariant under any rearrangement. This follows directly from the monotonicity of partial sums, which bound the series independently of order.^[28] Counterexamples illustrate the necessity of absolute convergence for full uniqueness: the alternating harmonic series

\sum (-1)^{n+1}/n

, which converges conditionally to

\ln 2

, can be rearranged to sum to any prescribed real number, underscoring that without absolute convergence, arbitrary rearrangements generally fail to preserve the sum.^[28]

Tests and Criteria

Dirichlet Test

The Dirichlet test provides a criterion for the convergence of infinite series of the form

\sum a_n b_n

, where the partial sums of

\sum b_n

are bounded and

a_n

decreases monotonically to zero. Specifically, let

\{B_n\}

denote the partial sums

B_n = \sum_{k=1}^n b_k

, and suppose there exists a constant

M > 0

such that

|B_n| \leq M

for all

n

. If the sequence

\{a_n\}

is monotonically decreasing and

\lim_{n \to \infty} a_n = 0

, then the series

\sum_{n=1}^\infty a_n b_n

converges.^[30]^[31] This test is particularly valuable for establishing conditional convergence, though it requires separate verification that

\sum |a_n b_n|

diverges to confirm the conditional nature.^[31] The proof relies on summation by parts, a discrete analog of integration by parts, to analyze the partial sums of

\sum a_n b_n

. Let

S_n = \sum_{k=1}^n a_k b_k

. Applying summation by parts yields

S_n = a_1 B_1 + \sum_{k=1}^{n-1} B_k (a_k - a_{k+1}) + a_n B_n.

Since

\{a_n\}

is monotonically decreasing to zero, the terms

a_k - a_{k+1} \geq 0

form a convergent series

\sum (a_k - a_{k+1})

, and the boundedness of

\{B_k\}

ensures that

\sum B_k (a_k - a_{k+1})

converges absolutely by comparison to

M \sum (a_k - a_{k+1})

. Additionally,

a_n B_n \to 0

n \to \infty

because

|a_n B_n| \leq M a_n \to 0

. Thus,

\{S_n\}

converges, implying the series

\sum a_n b_n

converges.^[30]^[31] A classic application is the alternating harmonic series

\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}

, which converges by the Dirichlet test with

a_n = 1/n

(monotonically decreasing to 0) and

b_n = (-1)^{n+1}

(partial sums

B_n

bounded by 1). The series

\sum |(-1)^{n+1}/n| = \sum 1/n

diverges, confirming conditional convergence.^[32]^[31] The test does not inherently distinguish between absolute and conditional convergence; while it guarantees convergence of

\sum a_n b_n

, one must apply other criteria (such as the comparison test) to

\sum |a_n b_n|

to determine if the convergence is conditional. It also fails if the partial sums of

\sum b_n

are unbounded or if

\{a_n\}

lacks monotonicity, even if both sequences tend to zero.^[30]^[32] Named after the German mathematician Peter Gustav Lejeune Dirichlet (1805–1859), the test originated in his foundational 1829 work on the convergence of Fourier series.^[33]

Abel's Test

Abel's test provides a sufficient condition for the convergence of the infinite series

\sum a_n b_n

, where

\{a_n\}

and

\{b_n\}

are sequences of real numbers. Specifically, if the series

\sum b_n

converges and the sequence

\{a_n\}

is monotonic with

\lim_{n \to \infty} a_n = 0

, then

\sum a_n b_n

converges.^[34] This test is a special case of the more general Dirichlet test, which requires only that the partial sums of

\sum b_n

are bounded rather than convergent. Since the convergence of

\sum b_n

implies that its partial sums are bounded, Abel's test follows directly as a corollary.^[34] The proof relies on summation by parts, analogous to integration by parts for integrals. Let

B_n = \sum_{k=1}^n b_k

, so

B_n

converges to some limit

B

n \to \infty

. For the partial sums of

\sum a_n b_n

, apply summation by parts:

\sum_{k=1}^n a_k b_k = a_n B_n - \sum_{k=1}^{n-1} B_k (a_{k+1} - a_k)

. Since

\{a_n\}

is monotonic and converges to 0, the differences

a_{k+1} - a_k

have consistent sign and bounded variation, and the boundedness of

B_k

ensures the remainder term converges, yielding convergence of the series. This argument derives immediately from the Dirichlet test by substituting the stronger condition on

\{B_n\}

.^[34] Abel's test originated in the early 19th century, with contributions from Niels Henrik Abel (1802–1829) in his work on rigorous approaches to infinite series, and was further developed and taught by Karl Weierstrass in his lectures on analysis.^[35] A representative example illustrating conditional convergence via Abel's test is the series

\sum_{n=1}^\infty \frac{(-1)^n}{n}

, the alternating harmonic series. Set

a_n = \frac{1}{\sqrt{n}}

an=n

1, which is monotonic decreasing to 0, and $b_n = \frac{(-1)^n}{\sqrt{n}}$

(−1)n. The series $\sum b_n$ converges by the alternating series test, since $\frac{1}{\sqrt{n}}$

1 decreases to 0. Thus, Abel's test implies $\sum a_n b_n = \sum \frac{(-1)^n}{n}$ converges. However, the absolute series $\sum \frac{1}{n}$ diverges by the integral test or p-series criterion with p=1, confirming conditional convergence.^[34] In extensions, Abel's test can establish conditional convergence when the absolute series $\sum |a_n b_n|$ diverges. For instance, if $\{a_n\}$ is positive and decreasing to 0, and $\sum |b_n|$ diverges while $\sum b_n$ converges (as in conditionally convergent cases), then comparison with the harmonic series or integral test often shows divergence of the absolute series, leveraging the test's conclusion for the original series.^[34]

Extensions and Applications

Improper Integrals

In the context of improper integrals, conditional convergence occurs when the integral

\int_a^\infty f(x) \, dx

converges as the limit

\lim_{b \to \infty} \int_a^b f(x) \, dx

exists and is finite, but the absolute integral

\int_a^\infty |f(x)| \, dx

diverges.^[36] This mirrors the notion for series but applies to continuous functions over unbounded domains, where cancellation of positive and negative contributions is essential for convergence without absolute integrability.^[36] A classic example is the Dirichlet integral

\int_0^\infty \frac{\sin x}{x} \, dx

, which converges to

\frac{\pi}{2}

.^[37] However, the absolute integral

\int_0^\infty \left| \frac{\sin x}{x} \right| \, dx

∫0∞

xsinx

dx diverges logarithmically, as the contributions over each interval $[n\pi, (n+1)\pi]$ behave like the harmonic series $\sum \frac{1}{n}$ . Criteria for conditional convergence of improper integrals are analogous to those for series. For instance, the Dirichlet test for integrals states that if $f$ is continuous on $[a, \infty)$ , the partial integrals $F(t) = \int_a^t f(x) \, dx$ are bounded for all $t \geq a$ , $g$ is positive, decreasing to 0 as $t \to \infty$ , and differentiable, then $\int_a^\infty f(x) g(x) \, dx$ converges.^[38] The proof relies on integration by parts: set $u = g(x)$ , $dv = f(x) \, dx$ , so $du = g'(x) \, dx$ , $v = F(x)$ ; then $\int_a^b f g \, dx = g(b) F(b) - g(a) F(a) - \int_a^b F(x) g'(x) \, dx$ . As $b \to \infty$ , $g(b) F(b) \to 0$ by boundedness of $F$ and $g \to 0$ , and the remaining integral converges absolutely since $|g'(x)| \leq -g'(x)$ (as $g' \leq 0$ ) and $\int_a^\infty -g'(x) \, dx = g(a) < \infty$ . Unlike conditionally convergent series, where rearrangements can alter the sum, improper integrals have a fixed order inherent to the continuum, making them less sensitive to discrete permutations but still dependent on the choice of limits, such as in principal value senses where symmetric exclusions around singularities ensure convergence.^[39]

Multidimensional Series

In the context of multidimensional series, conditional convergence extends the univariate notion to series with multiple indices, such as double series

\sum_{m=1}^\infty \sum_{n=1}^\infty a_{m,n}

. A double series converges in the Pringsheim sense if the rectangular partial sums

S_{m,n} = \sum_{i=1}^m \sum_{j=1}^n a_{i,j}

approach a finite limit as

m, n \to \infty

independently of how

m

and

n

tend to infinity. The series is conditionally convergent if it converges in this sense but the absolute double series

\sum_{m=1}^\infty \sum_{n=1}^\infty |a_{m,n}|

diverges.^[40] A representative example is the double series

\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{(-1)^{m+n}}{m n}

. The iterated sums converge because the inner sum over

n

for fixed

m

is the alternating harmonic series scaled by

(-1)^m / m

, and similarly for the other iteration, yielding a finite value of

(\ln 2)^2

. However, the absolute series

\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m n}

diverges as the product of two divergent harmonic series. Thus, the series converges conditionally. The Steinitz theorem addresses the complexities of order dependence in finite-dimensional spaces. For a conditionally convergent series

\sum_{k=1}^\infty \mathbf{x}_k

\mathbb{R}^d

, the set of all possible sums obtained by rearrangements forms an affine subspace: the sum under one convergent rearrangement plus the orthogonal complement to the subspace of linear functionals

\Gamma = \{ f \in (\mathbb{R}^d)^* : \sum_{k=1}^\infty |f(\mathbf{x}_k)| < \infty \}

. This result shows that absolute convergence (

\sum_{k=1}^\infty \|\mathbf{x}_k\| < \infty

) implies convergence independent of the summation order, while conditional convergence permits sums to vary within that affine subspace depending on the order.^[41] Multidimensional series introduce additional challenges beyond univariate cases, particularly in defining convergence. Pringsheim convergence relies on rectangular partial sums, whereas Cesàro summation uses averages of those sums over rectangular regions; a series may converge in the Cesàro sense but diverge in the Pringsheim sense, affecting assessments of conditional convergence. Other variants include square convergence (along diagonals

m + n = k

) versus rectangular, and iterated convergence (summing rows then columns, or vice versa), where the double series may fail to converge even if iterated sums do, or the limit may depend on the iteration order. These discrepancies highlight the sensitivity of conditional convergence to summation methods in higher dimensions.^[42]^[40] In applications, conditional convergence of multidimensional series arises in Fourier analysis for functions of several variables, where multiple Fourier series

\sum_{m,n \in \mathbb{Z}} \hat{f}(m,n) e^{i (m x + n y)}

may converge pointwise or in other senses to the function under conditions like bounded variation or Lipschitz continuity, but absolute convergence requires stricter assumptions such as higher-order smoothness. This extends the univariate Riemann rearrangement theorem to multiple indices, with parallels to improper integrals as a continuous analog.^[43]

Info Pages

Talk Pages

Special Pages

Conditional convergence

Recent from talks

Recent from talks

Contribute something

Contribute something

Media Pages

Timelines

Articles

Notes collections

Notes

Notes

Days in Chronicle

Conditional convergence

Definition

See also

References

Conditional convergence

Convergence Basics

Absolute Convergence

Conditional Convergence

Examples and Illustrations

Alternating Harmonic Series

Other Standard Examples

Theoretical Properties

Riemann Rearrangement Theorem

Uniqueness of Sums Under Rearrangement

Tests and Criteria

Dirichlet Test

Abel's Test

Extensions and Applications

Improper Integrals

Multidimensional Series

References

Add your contribution

Related Hubs

Contribute something

History

Conditional convergence

Recent from talks

Recent from talks

Contribute something

Contribute something

Media Pages

Timelines

Articles

Notes collections

Notes

Notes

Days in Chronicle

Conditional convergence

Definition

See also

References

Conditional convergence

Convergence Basics

Absolute Convergence

Conditional Convergence

Examples and Illustrations

Alternating Harmonic Series

Other Standard Examples

Extensions and Applications

Improper Integrals

Add your contribution

Related Hubs

Contribute something