Finding Interval of Convergence

Calculus 2 · Axiom Academy

LESSON Finding the Interval of Convergence A power series converges on a band around its center — find its radius with the ratio test, then check each endpoint by hand. For a power series centered at a , the same three moves find exactly where it converges. The radius gives the open interior; the endpoints are decided one at a time; the brackets record the verdicts. 2. Step 1 — Apply the Ratio Test The ratio test compares consecutive terms. For a power series the coefficient ratio settles to a number, and the leftover |x-a| rides along. The series converges absolutely whenever that limit is below 1 . Solving for |x-a| gives a radius R . Inside it the series definitely converges; the open interval is locked in. 3. Step 2 — Check the Endpoints At an endpoint the ratio test gives L = 1 — inconclusive . So plug each endpoint back into the series and judge the resulting numerical series on its own. The two endpoints can disagree, so you must check both. Alternating Series Test — terms alternate sign and shrink to 0 . p -series — compare to ( converges). Integral test — for positive, decreasing terms. Direct comparison — against a known series. 4. Step 3 — Write the Interval The two endpoint verdicts pick the brackets. A square bracket means the endpoint is included (it converged); a round bracket means it is excluded (it diverged). Four combinations are possible. Run all three steps on one series and watch the interval assemble itself.

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