Rearrangement of Series
Conditionally convergent series can be rearranged to any sum (Riemann).
Key idea
Conditionally convergent series can be rearranged to any sum (Riemann).
In depth
For absolutely convergent series, rearranging terms does not change the sum. For conditionally convergent series, Riemann’s theorem says rearrangements can produce any real sum, or diverge.
Worked example
The alternating harmonic series 1 − 1/2 + 1/3 − ... = ln 2 can be rearranged to sum to (3/2) ln 2.
Real-world uses
- Explaining why summing an infinite series in a different order can change its answer
Common mistake
Reordering terms of a conditionally convergent series freely.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Rearrangement of Series?
Conditionally convergent series can be rearranged to any sum (Riemann). It is a topic at the expert level in Analysis & Topology.
Where is Rearrangement of Series used in real life?
Real-world examples: Explaining why summing an infinite series in a different order can change its answer.
What should I learn before Rearrangement of Series?
Make sure you are comfortable with Series Convergence (Analysis), the previous topic in Analysis & Topology.
What is a common mistake with Rearrangement of Series?
Reordering terms of a conditionally convergent series freely.
How can I practise Rearrangement of Series?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Rearrangement of Series?
Move on to Limits of Functions (ε–δ) next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.