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Sequences & Series · Power & Taylor Series

Power Series

∑n=0∞cn(x−a)n\sum_{n=0}^{\infty} c_n (x - a)^n

A power series centered at a. It converges for |x - a| < R (radius of convergence) and diverges for |x - a| > R. Check endpoints separately.

Worked examples

Find the radius of convergence of Σ xⁿ/n.
  1. Ratio test: |a_{n+1}/aₙ| = |x|·n/(n+1) → |x|
  2. Converges when |x| < 1. R = 1.
  3. At x=1: Σ 1/n diverges. At x=-1: Σ(-1)ⁿ/n converges.

Answer: R = 1, interval of convergence: [-1, 1).

Related formulas

Radius of Convergence

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