Lecture 22: Absolute Convergence and
Ratio and Root Tests (II)
The Root Test-
Let
X
j
a
n
j
be a series.
1. If lim
n
!1
n
p
j
a
n
j
=
L <
1
;
then the series
is absolutely convergent.
2. If lim
n
!1
n
p
j
a
n
j
=
L >
1
;
or lim
n
!1
n
p
j
a
n
j
=
1
,
then the series is divergent.
3. If lim
n
!1
n
p
j
a
n
j
= 1
;
then the
Root Test
is inconclusive.
1

ex.
1
X
n
=1
n
2
n
+ 5
n
ex.
1
X
n
=1
1 +
1
n
n
2
2

Remark:
lim
n
!1
n
p
n
= 1
ex.
1
X
n
=2
n
(ln
n
)
n
3

ex.
1
X
n
=1
cos(
1
n
)
n
1
n
ex.
1
X
n
=1
e
2
n
n
n
4

ex.
1
X
n
=1
(2
n
)!
(
n
!)
2
5

ex.
1
X
n
=1
1
3
5
(2
n
1)
(2
n
)!
6

NYTI:
1
X
n
=1
sin(
1
n
)
n
(conv: LCT with
b
n
=
1
n
2
)
1
X
n
=1
(cos
x
)
2
n
2
n
(conv: geometric
r
=
cos
2
x
2
)
1
X
n
=3
(
1)
n
ln
n
(conditional convergent)
1
X
n
=1
n
!
n
n
(conv: Ratio ,
1
e
<
1 )
1
X
n
=1
n
+ 1
n
n
2
(div,root)
Absolute or Conditional convergent?
1
X
n
=1
cos(
n =
3)
n
!
(Conv:
P
1
n
!
con,Ratio ;
j
a
n
j
=
j
cos(
n
=
3)
j
n
!
1
n
!
, hence the
original series converges Absolutelyby Comparison test, that implies the original series con-
verges.)
7