WORKSHEET 07/12/2016
1. Determine which of the following series absolutely converge, conditionally converge or
diverge :
p
P1
P1
P1
n+1 p1
n 3p n+1
n 1
(
1)
n=1 ( 1)
n=1
n=1 ( 1) ln n
n
n+1
P1
n=1 (
P1
n=1 (
n
10
1)n (n+1)!
P1
n=1 (
1)n+1 (0.1)
n
P1
n
n=1 (
1)n+1 2n!n
1)n arctan(n)
(n2 +1)
2. a) Find the series’ radius and interval of convergence. For what values of x does the
series converge b) absolutely c) conditionally:
P1 (3x 2)n
P1 ( 1)n xn
P1 4n x2n
n=1
P1
n=1
n
n(x+3)n
5n
n=1
P1
n=1 (1
n=1
n!
+ n1 )n xn
1
n
P1
n=1 n!(x
4)n
3. The series
sec x = 1 +
x2
2
+
5 4
24 x
+
61 6
720 x
+
277 8
8064 x
+ ...
converges to sec x for ⇡/2 < x < ⇡/2 .
a. Find the first five terms of a power series for the function ln | sec x + tan x|. For
what values of x should the series converge?
b. Find the first four terms of a series for sec x tan x . For what values of x should the
series converge?
2