Page 267 - Calculus Workbook For Dummies

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Chapter 13: Infinite Series: Welcome to the Outer Limits

Ratiocinating the Two “R” Tests

Here you practice the ratio test and the root test. With both tests, a result less than 1
means that the series in question converges; a result greater than 1 means that the
series diverges; and a result of 1 tells you nothing.

The ratio test: Given a series ! , consider the limit of the ratio of a term to the
u n
previous term, lim u n 1+ . If this limit is less than 1, the series converges. If it’s
n " 3 u n
greater than 1 (this includes 3), the series diverges. And if it equals 1, the ratio
test tells you nothing.
The root test: Note its similarity to the ratio test. Given a series ! , consider
u n
the limit of the nth root of the nth term, lim n u n . If this limit is less than 1, the
n " 3
series converges. If it’s greater (including 3), the series diverges. And if it equals
1, the root test says nothing.

The ratio test is a good test to try if the series involves factorials like !n or where n is
n
in the power like 2 . The root test also works well when the series contains nth
powers. If you’re not sure which test to try first, start with the ratio test — it’s often
the easier to use.
Sometimes it’s useful to have an idea about the convergence or divergence of a series
before using one of the tests to prove convergence or divergence.

3
3
Q. Does ! n n converge or diverge? Q. Does ! 5 3 n + n converge or diverge?
4
n 1 2 n 1 n 3
=
=
A. Try the ratio test. A. Consider the limit of the nth root of the nth
term:
n + 1 n
2 (n 1 ) 2 ^ n + 1h n + 1 1 5 3 n + 4 5 3 n + 4 1 /n 5 3 + 4 /n
+
lim n = lim n 1 = lim = lim n n = limd n n = lim 3 = 0
+
n " 3 n n " 3 2 : n n " 3 2 n 2 n " 3 n 3 n " 3 n 3 n " 3 n
2
Because this is less than 1, the series Because this limit is less than 1, the
converges. series converges.