Interval for Σ(xⁿ/n)

Calculus 2 · Axiom Academy

Use the Ratio Test and endpoint analysis to find where this power series converges. Find the interval of convergence of the power series . You found the interval of convergence by combining the Ratio Test with a separate check at each endpoint. Ratio Test first: Start by applying the Ratio Test to find the radius of convergence R . The series converges for . Check endpoints separately: The Ratio Test is inconclusive at the endpoints (where L = 1 ), so test x = c - R and x = c + R individually. Different tests for endpoints: At x = -1 we got an alternating series (Alternating Series Test). At x = 1 we got a p -series with p = 1 — the harmonic series, which diverges. Interval notation matters: Square brackets for included endpoints, parentheses for excluded ones. Our answer [-1, 1) includes -1 but excludes 1 . This approach works for any power series: use the Ratio Test for the radius, then carefully check each endpoint. A series may converge at both, one, or neither endpoint.

This is the written version of the interactive lesson above. See the full Calculus 2 course.