136 4. SEQUENCES, SERIES, AND FUNCTION APPROXIMATION
Knowledge Box 4.15
e ratio test
Suppose that
1
X
nD0
x
n
is a series. Compute
r D lim
n!1
ˇ
ˇ
ˇ
ˇ
x
nC1
x
n
ˇ
ˇ
ˇ
ˇ
:
en:
• if r < 1, then the series converges,
• if r > 1, then the series diverges,
• if r D 1, then the test is inconclusive.
Definition 4.10 e quantity nŠ D n .n 1/ .n 2/ 2 1 is called n factorial.
We define 0Š D 1.
Example 4.32 Determine if
1
X
nD0
1
nŠ
converges or diverges.
Solution:
Use the ratio test.
lim
n!1
ˇ
ˇ
ˇ
ˇ
1=.n C1/Š
1=nŠ
ˇ
ˇ
ˇ
ˇ
D lim
n!1
ˇ
ˇ
ˇ
ˇ
nŠ
.n C1/Š
ˇ
ˇ
ˇ
ˇ
D lim
n!1
n.n 1/ 2 1
.n C1/n.n 1/ 2 1
D lim
n!1
1
n C1
D 0
Since 0 < 1, we can deduce that the series converges by the ratio test.
˙
4.2. SERIES CONVERGENCE TESTS 137
Knowledge Box 4.16
e root test
Suppose that
1
X
nD0
x
n
is a series. Compute
s D lim
n!1
n
p
jx
n
j:
en:
• if s < 1, then the series converges,
• if s > 1, then the series diverges,
• if s D 1, then the test is inconclusive.
Example 4.33 Determine if
1
X
nD0
1
n
n
converges or diverges.
Solution:
Use the root test.
lim
n!1
n
s
ˇ
ˇ
ˇ
ˇ
1
n
n
ˇ
ˇ
ˇ
ˇ
D lim
n!1
1
n
D 0
Since 0 < 1, we can deduce that the series converges by the root test.
˙
e root test and, especially, the ratio test will get a big workout in the next section. is section
contains nine tests for series convergence – many of which depend on knowing the convergence
or divergence behavior of other series. is creates a mental space very similar to the “which
integration method do I use?” issue that arose in Fast Start Integral Calculus. e method for
dealing with this is the same here as it was there: practice, practice, practice.
It’s also good to keep in mind, when you are searching for series to compare to, that the examples
in this section may be used as examples for comparison. A list of series with known behaviors is
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.