Assign8 - Math 128 ASSIGNMENT 8 Winter 2009 Submit all...

Math 128 ASSIGNMENT 8 Winter 2009
Submit all problems by 8:20 am on Wednesday, March 25 th in the drop boxes across
from MC4066, or in class depending on your instructor’s preference.
All solutions must be clearly stated and fully justified .
Comment: This assignment is longer than usual. It is strongly recommended that you start
working on it early.
1. Find the sum of the given geometric series.
a)
∞
n =0
3
(
-
1
4
)
n
b)
∞
n =0
π
n
2 3 n - 1
c)
∞
n =0
3 + 2
n
3 n +2
2. Apply the n th Term Test (i.e. Divergence Test) or the Comparison Test to decide
whether each series converges, converges absolutely, or diverges.
a)
∞
n =2
n + 1
n - 1
b)
∞
n =1
sin nx
2 n
, x ∈ R c)
∞
n =1
2
n
n 3 n
d)
∞
n =0
1
1 + 3 n
e)
∞
n =1
1
2
-
1
n 2
f)
∞
n =0
cos( n 2 )
π n + 3
3. Determine whether each of the following series converges absolutely or diverges by
comparison to an appropriate p -series. (You may use the result of example 2 on page
699 of your text.)
(i)
∞
n =2
1
√
n - 1
(ii)
∞
n =1
( - 1) n n
n 4 + 2
(iii)
∞
n =1
arctan 2 n
n 2
4. Apply the integral test to determine whether or not each series converges.