## The Root Test

The root test gives information about the convergence or not of a series by considering Theorem (The Root Test).

If there exists some N such that for some q with n&gt;=N thenc onverges absolutely. If for infinitely many then diverges.

Proof: Suppose there is a real number such that and a positive number such that for  then and since converges, hence by the comparison test converges absolutely.

Suppose now that for infinitely many then for infinitely many so does not converge to zero and does not converge.

Example: even, odd. Even and odd. Thus hence by the root test the series converges absolutely.

Example: Does converge for  so  converges to 1 so converges to Let then so there is a positive integer such that implies that thus converges absolutely.

The root test has wider scope than the ratio test, since if the root test gives no results for convergence, then neither does the ratio test. Sometimes however the root test can be hard to use, and more information can usefully be obtained from the ratio test. Neither root nor ratio test will identify series that converge conditionally. 