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.
- 0 variables
- 2 worked examples
- 18 in Sequences & Series
Worked examples
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).
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Find the interval of convergence of Σ (x-2)ⁿ/(n·3ⁿ).
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