## The Root Test

Since and are inverses, we have . This implies the very useful fact that

As a sample application, notice that for any nonzero ,

Similarly,

where we've used that , which we could prove using L'Hopital's rule.

Theorem 6.4.10 (Root Test)   Consider the sum .
1. If , then convergest absolutely.
2. If , then diverges.
3. If , then we may conclude nothing from this!

Proof. We apply the comparison test (Theorem 6.4.1). First suppose . Then there is a such that for we have . Thus for such we have . The geometric series converges, so also does, by Theorem 6.4.1. If for , then we see that diverges by comparing with .

Example 6.4.11   Let's apply the root test to

We have

Thus the root test tells us exactly what we already know about convergence of the geometry series (except when ).

Example 6.4.12   The sum is a candidate for the root test. We have

Thus the series converges.

Example 6.4.13   The sum is a candidate for the root test. We have

hence the series diverges!

Example 6.4.14   Consider . We have

so we conclude nothing!

Example 6.4.15   Consider . To apply the root test, we compute

Again, the limit diverges, as in Example 6.4.8.

William Stein 2006-03-15