Weierstrass M-Test
Test for uniform convergence of series.
Key idea
Test for uniform convergence of series.
In depth
If |fₙ(x)| ≤ Mₙ for all x and ΣMₙ converges, then Σfₙ converges uniformly (and absolutely). It is the main practical test for uniform convergence of series.
Worked example
Σ sin(nx)/n² converges uniformly on R, since |sin(nx)/n²| ≤ 1/n².
Real-world uses
- Proving that a series of functions converges uniformly, used in Fourier and power series analysis
Common mistake
Using bounds Mₙ that depend on x.
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 Weierstrass M-Test?
Test for uniform convergence of series. It is a topic at the expert level in Analysis & Topology.
Where is Weierstrass M-Test used in real life?
Real-world examples: Proving that a series of functions converges uniformly, used in Fourier and power series analysis.
What should I learn before Weierstrass M-Test?
Make sure you are comfortable with Pointwise vs Uniform Convergence, the previous topic in Analysis & Topology.
What is a common mistake with Weierstrass M-Test?
Using bounds Mₙ that depend on x.
How can I practise Weierstrass M-Test?
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 Weierstrass M-Test?
Move on to Uniform Limits and Continuity 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.