468 Chapter 8 • Infinite Series of Real Numbers
THE ROOT TEST
Theorem 8.2.14 (Root Test, Basic Form) Let I: an be a nonnegative
series.
(a) If ::3 0 < r < 1 3 ~ < r for all but finitely many n, then L::an
converges.
(b) If ~ 2:: 1 for infinitely many n, then I: an diverges.
Proof. Let I: an be a nonnegative series.
(a) Suppose ::3 0 < r < 1 3 ~::::; r for all but finitely many n. That is,
::3 n 0 EN 3 n 2:: n 0 => ~::::; r. Then n 2:: n 0 =>an::::; rn. Since 0 < r < 1, the
geometric series I: rn converges. Thus, by the comparison test, I: an converges.
(b) Suppose ~ ;::: 1 for infinitely many n. Then an 2:: 1 for infinitely
many n, so an ft o. Thus, by the general term test, Lan diverges. •
The root test may be more familiar to you in its limit form, which can
easily be proved as a coroll ary to Theorem 8.2.14.
Corollary 8.2.15 (Root Test, Limit Form) If I: an is a nonnegative se-
ries, and n-+oo lim ~ = R (possibly +oo) then
(a) if R < 1, the series I: an converges, and
(b) if R > 1, the series I: an diverges.
Proof. Exercise 38. •
As with the ratio test, the limit form of the root test cannot be used in
cases where lim ~fails to exist. However, this can be overcome by using
n-+oo
the upper limit, which is guaranteed to exist (but may be infinite).
Theorem 8.2.16 (Root Test, Upper Limit Form)^2 Suppose Lan is a
nonnegative series and let R = lim ~ (possibly +oo). Then
n-+oo
(a) ifR<l, theseriesL:an converges, and
(b) if R > 1, the series I: an diverges.
Proof. Exercise 39. •
- Compare with Theorem 8.2.13, in which both upper and lower limits are required.