Student number
Name [SURNAME(S), Givenname(s)]
MATH 101, Section 212 (CSP)
Week 8: Marked Homework Assignment
Due: Thu 2011 Mar 10 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Find the limit for each the following sequences, or else determine that the sequence is
divergent.
(a) limn→∞
2n+1
√
1−3 n
(b) limn→∞
n+(−1)n+1
n
(c) limn→∞ [1 + (−1)n ]
sin n
n
n!
limn→∞ 106n
(d) limn→∞
(e)
2. Let a1 is given, and let an+1 = 4an (1 − an ) for n = 1, 2, 3, . . .. Find all values of a1
such that {an } is a constant sequence (every an is the same value).
√
3. (a) If a is positive and a < 2, prove that a < 2a.
√
(b) If a is positive and a < 2, prove that 2a < 2.
√
√
(c) Let a1 = 2 and let an+1 = 2an for n = 1, 2, 3, . . .. Prove that the sequence
converges.
(d) Find limn→∞ an .
4. For each of the following series, determine if it is convergent or divergent. If it is
convergent, find the sum.
(a)
P∞
(b)
P∞
(c)
n=0
1
2n
+
(−1)n
5n
1
n=1 cos n
P∞
6n+3
n=1 n2 (n+1)2
5. Express 1.24123 = 1.24123123123 . . . as a ratio of integers.
6. For each of the following series, find the values of x for which the series converges, and
find the sum of the series for those values of x.
(a)
(b)
P∞
n=1
P∞
n=0
2 n xn
sinn x