160 CHAPTER 6. INFINITE SERIES
To verify the divergence part, note that if r > 1, then 6.6 can be turned around for someR > 1. Showing limn→∞ |an| = ∞. Since the nth term fails to converge to 0, it follows theseries diverges.
To see the test fails if r = 1, consider ∑n−1 and ∑n−2. The first series diverges whilethe second one converges but in both cases, r = 1. (Be sure to check this last claim.)
The ratio test is very useful for many different examples but it is somewhat unsatisfac-tory mathematically. One reason for this is the assumption that an ̸= 0, necessitated by theneed to divide by an, and the other reason is the possibility that the limit might not exist.The next test, called the root test removes both of these objections.
Theorem 6.3.2 Suppose |an|1/n < R < 1 for all n sufficiently large. Then ∑∞n=1 an
converges absolutely. If there are infinitely many values of n such that |an|1/n ≥ 1, then∑
∞n=1 an diverges.
Proof: Suppose first that |an|1/n < R < 1 for all n sufficiently large. Say this holdsfor all n ≥ nR. Then for such n, n
√|an| < R.Therefore, for such n, |an| ≤ Rn and so the
comparison test with a geometric series applies and gives absolute convergence as claimed.Next suppose |an|1/n ≥ 1 for infinitely many values of n. Then for those values of n,
|an| ≥ 1 and so the series fails to converge by the nth term test, Theorem 6.2.13.Stated more succinctly, using Definition 3.3.16 the condition for the root test is this:
Let r ≡ limsupn→∞ |an|1/n then
∞
∑k=m
ak
converges absolutely if r < 1
test fails if r = 1
diverges if r > 1
To see the test fails when r = 1, consider the same example given above, ∑n1n and ∑n
1n2 .
A special case occurs when the limit exists.
Corollary 6.3.3 Suppose limn→∞ |an|1/n exists and equals r. Then
∞
∑k=m
ak
converges absolutely if r < 1
test fails if r = 1
diverges if r > 1
Proof: The first and last alternatives follow from Theorem 6.3.2. To see that the testfails if r = 1, consider the two series ∑
∞n=1
1n and ∑
∞n=1
1n2 both of which have r = 1 but
having different convergence properties. The first diverges and the second converges.
6.4 Exercises1. Determine whether the following series converge and give reasons for your answers.