Riemann series theorem

Key takeaways

Riemann series theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Riemann series theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Riemann series theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent. As an example, the series

1 − 1 + 1 2 − 1 2 + 1 3 − 1 3 + 1 4 − 1 4 + … {\displaystyle 1-1+{\frac {1}{2}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{4}}+\dots }

converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values gives

1 + 1 + 1 2 + 1 2 + 1 3 + 1 3 + … {\displaystyle 1+1+{\frac {1}{2}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{3}}+\dots }

which sums to infinity. Thus, the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the next two positive terms and then the next negative term, etc.) to give a series that converges to a different sum, such as

1 + 1 2 − 1 + 1 3 + 1 4 − 1 2 + … {\displaystyle 1+{\frac {1}{2}}-1+{\frac {1}{3}}+{\frac {1}{4}}-{\frac {1}{2}}+\dots }

which evaluates to ln 2. More generally, using this procedure with p positives followed by q negatives gives the sum ln(p/q). Other rearrangements give other finite sums or do not converge to any sum.

History It is a basic result that the sum of finitely many numbers does not depend on the order in which they are added. For example, 2 + 6 + 7 = 7 + 2 + 6. The observation that the sum of an infinite sequence of numbers can depend on the ordering of the summands is commonly attributed to Augustin-Louis Cauchy in 1833. He analyzed the alternating harmonic series, showing that certain rearrangements of its summands result in different limits. Around the same time, Peter Gustav Lejeune Dirichlet highlighted that such phenomena are ruled out in the context of absolute convergence, and gave further examples of Cauchy's phenomenon for some other series which fail to be absolutely convergent. In the course of his analysis of Fourier series and the theory of Riemann integration, Bernhard Riemann gave a full characterization of the rearrangement phenomena. He proved that in the case of a convergent series which does not converge absolutely (known as conditional convergence), rearrangements can be found so that the new series converges to any arbitrarily prescribed real number. Riemann's theorem is now considered as a basic part of the field of mathematical analysis. For any series, one may consider the set of all possible sums, corresponding to all possible rearrangements of the summands. Riemann's theorem can be formulated as saying that, for a series of real numbers, this set is either empty, a single point (in the case of absolute convergence), or the entire real number line (in the case of conditional convergence). In this formulation, Riemann's theorem was extended by Paul Lévy and Ernst Steinitz to series whose summands are complex numbers or, even more generally, elements of a finite-dimensional real vector space. They proved that the set of possible sums forms a real affine subspace. Extensions of the Lévy–Steinitz theorem to series in infinite-dimensional spaces have been considered by a number of authors.

Definitions A series ∑ n = 1 ∞ a n {\textstyle \sum _{n=1}^{\infty }a_{n}} converges if there exists a value ℓ {\displaystyle \ell } such that the sequence of the partial sums

( S 1 , S 2 , S 3 , … ) , S n = ∑ k = 1 n a k , {\displaystyle (S_{1},S_{2},S_{3},\ldots ),\quad S_{n}=\sum _{k=1}^{n}a_{k},}

converges to ℓ {\displaystyle \ell } . That is, for any ε > 0, there exists an integer N such that if n ≥ N, then

| S n − ℓ | ≤ ε . {\displaystyle \left\vert S_{n}-\ell \right\vert \leq \varepsilon .}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Riemann series theorem

Start with the simplest possible case. Write down what Riemann series theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Riemann series theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Riemann series theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Riemann series theorem

In research: Riemann series theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Riemann series theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Riemann series theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Bernhard Riemann, Permutations, Series (mathematics), so understanding it makes those chapters shorter.
In everyday life: Look for Riemann series theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Riemann series theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Riemann series theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Riemann series theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Riemann series theorem in simple terms?

In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new ser…

Why does Riemann series theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Riemann series theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Riemann series theorem.