Power Series Radius of Convergence
Cauchy–Hadamard formula.
Key idea
Cauchy–Hadamard formula.
In depth
For Σaₙxⁿ, R = 1/limsup|aₙ|^(1/n). The series converges absolutely for |x| < R and diverges for |x| > R. Inside the radius it can be differentiated and integrated term by term.
Worked example
Σ xⁿ/2ⁿ has R = 2, converging for |x| < 2.
Formula
Real-world uses
- Finding where a series approximation for a function is valid in physics and engineering
Common mistake
Forgetting to test the endpoints separately.
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 Power Series Radius of Convergence?
Cauchy–Hadamard formula. It is a topic at the expert level in Analysis & Topology.
What is the formula for Power Series Radius of Convergence?
The key formula is 1/R = limsup |aₙ|^(1/n).
Where is Power Series Radius of Convergence used in real life?
Real-world examples: Finding where a series approximation for a function is valid in physics and engineering.
What should I learn before Power Series Radius of Convergence?
Make sure you are comfortable with Uniform Limits and Continuity, the previous topic in Analysis & Topology.
What is a common mistake with Power Series Radius of Convergence?
Forgetting to test the endpoints separately.
How can I practise Power Series Radius of Convergence?
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 Power Series Radius of Convergence?
Move on to Taylor’s Theorem with Remainder 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.