Sequences & Series · Power & Taylor Series
Power Series
A power series centered at a. It converges for |x - a| < R (radius of convergence) and diverges for |x - a| > R. Check endpoints separately.
- 2 worked examples
- 18 in Sequences & Series
Worked examples
- Ratio test: |a_{n+1}/aₙ| = |x|·n/(n+1) → |x|
- Converges when |x| < 1. R = 1.
- At x=1: Σ 1/n diverges. At x=-1: Σ(-1)ⁿ/n converges.
Answer: R = 1, interval of convergence: [-1, 1).
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Common questions
What is the Power Series?
A power series centered at a. It converges for |x - a| < R (radius of convergence) and diverges for |x - a| > R. Check endpoints separately. It is one of the sequences & series formulas in the CalcRef reference.
How do you use the Power Series?
Worked example. Find the radius of convergence of Σ xⁿ/n. Ratio test: |a_{n+1}/aₙ| = |x|·n/(n+1) → |x|. Converges when |x| < 1. R = 1. At x=1: Σ 1/n diverges. At x=-1: Σ(-1)ⁿ/n converges. Answer: R = 1, interval of convergence: [-1, 1).
Practice
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