MAT 137
Tutorial #18– Convergence Tests
March 20-21, 2017
Determine which ones of the following series are absolutely convergent, conditionally convergent,
or divergent.
1
X
e1 1/n
1.
3 + sin n
n=1
1
X
1
2.
n
n=1
3.
1
X
n=1
4.
5.
6.
12.
( 1)n
n
1
X
( 1)n
n2 + 6
n=1
n=1
n2
p
n5 + 4n + 11
1
X
1
7.
( 1)n sin
n
n=1
1
X
1
8.
( 1) n sin
n
n=1
9.
10.
n
1
X
(n + 3)2n
n=1
n!
1
X
n!(2n)!
n=1
1
X
n=1
1
X
1
n2
n=1
1
X
1
X
1
11.
nn
n=1
(3n)!
13.
2 · 4 · 6 · . . . · (2n)
⇡ n+1
· 2n 1
1 · 3 · 5 · . . . · (2n 1) e
1
X
sin n
n=1
n2
1
X
n!
14.
104n
n=1
15.
16.
1
2
+ 2
2 3
1
X
cos(n⇡)
n=2
17.
1
X
n=2
18.
19.
ln n
1
n (ln n)2
1
X
1
ln n
n=2
1
X
ln n
n=2
20.
4
8
16
+ 4+ 5
3
4
5
6
1
X
n=2
n1.1
1
(ln n)3
32
+ ...
76