Discrete Mathematics Demystified

296 Discrete Mathematics Demystified
provided that j ≥ 2. Thus the series from Example 13.45 is termwise dominated
by the series
∞
j=1
Thus convergence can also be obtained from the comparison test. The root test,
however, proves to be more straightforward to apply in this example.
EXAMPLE 13.46
Apply the root test to the series
∞
j=1
1
j 3
1
[ln( j + 2)] j
Solution: Observe that c j = 1/[ln( j + 2)] j . Thus
|c j| 1/ j =
1
ln( j + 2)
This expression tends to 0 as j →∞. Therefore L = 0 < 1 and the Root test says
that the series converges absolutely. (Try using the ratio test on this one! How about
the Comparison Test?)
You Try It: Discuss convergence or divergence for the series
j 2 j /(4 j + 3 j ).
EXAMPLE 13.47
Apply the root test to the series
∞
j=1
2 j
j 10
Solution: We have c j = 2 j / j 10 . Therefore
|c j| 1/ j 2
=
( j 1/ j ) 10
(13.16)
Since j 1/ j → 1as j →∞, the expression (13.16) tends to 2 = L > 1. We conclude,
using the root test, that the series diverges.