The Root Test

Calculus 2 · Axiom Academy

A convergence test built for series with nth powers — take the nth root, and the exponent melts away. For a series , form the limit of the n th root of the absolute value of each term. Where that limit L lands relative to 1 is the whole verdict. the series converges absolutely . L = 1 the test is inconclusive — it tells you nothing; try another test. the series diverges (this includes ). Consider . The general term is already raised to the n th power, so the Root Test is the natural tool. Taking the n th root cancels that exponent cleanly: So the whole question reduces to one easy limit. Watch the sequence build, term by term, and settle onto its limit: The Root Test is most effective when a_n has an n in the exponent. Then simplifies and usually lands away from 1 — a clean verdict. When there is no n th power, the root often drifts back to L = 1 and tells you nothing. Whole term raised to the n th power: gives — well below 1 , so it converges . Any where n sits in the exponent. A p -series like has no n th power: , so L = 1 and the test is silent. The Ratio Test is no help here either — , so it is inconclusive too. For a p -series, use the p -series test ( , so it converges ) or the Integral Test. 4. Its Bond with the Ratio Test The Root and Ratio Tests are deeply linked. There is a theorem: if exists, then exists too and equals it. Whenever the Ratio Test gives a conclusive answer, the Root Test gives the same one.

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