A08 - terms prove that s-s 6 ≤ 1 729 4 Determine whether...

Math 138 – Fall 2011
Assignment 8
Due Monday Nov 14 (in the drop box by 12 noon)
Hand in the following:
1. Determine whether or not the following series converge or diverge by
making a suitable comparison.
(a)
∞
X
n =1
√
n
n 2 + 2 n + 2
(b)
∞
X
n =1
sin
1
n
(c)
∞ X
n =1
3 1 /n
n
3
(d)
∞ X
n =1
arctan n
n
(e)
∞
X
n =1
1 +
1
n
e n
(f)
∞
X
n =1
1
n
1+
1
n
2. (a) Show that
ln x
x 1 / 3
→ 0 as x → ∞ .
(
this is not a ‘prove using the definition’ question...just a simple calculate the limit. You
may want to review L’Hopital’s rule in Section 4.4 of the text.)
(b) From part (a) we can thus say that the sequence
ln n
n 1 / 3
∞
1
con-
verges to 0. Use the definition of convergence with = 1 , to
deduce that for sufficiently large n , (ln n )
3
≤ n .

(c) Show that the series
∞ X
n =2
1
(ln n )
3
diverges.
3. (a) By making a comparison to a certain geometric series, show that
∞
X
n =1
1 + sin n
1 + 3 n
converges.
(b) If s is the sum of the above series and s
6
is the sum of the first 6