Math 3210-1
HW 10
Due Tuesday, July 3, 2007
NOTE: Only turn in problems 1(d), 2(a), 2(c), 3(c), 4(a), and 6-9.
Sequences
1. Write out the first seven terms of each sequence.
(a) an = n2
(−1)n
(b) bn =
n
(c) cn = cos nπ
3
2n + 1
(d) dn =
3n − 1
2. Using only the definition of a limit of a sequence, prove the following.
k
(a) For any real number k, lim
= 0.
n→∞ n
1
(b) For any real number k > 0, lim
= 0.
n→∞ nk
3n + 1
= 3.
(c) lim
n→∞ n + 2
sin n
(d) lim
= 0.
n→∞ n
n+2
= 0.
(e) lim 2
n→∞ n − 3
3. Using any of the Theorems 47-49 or the examples we worked in class from section 4.1, prove the
following.
1
= 0.
1 + 3n
4n2 − 7
=0
lim
n→∞ 2n3 − 5
6n2 + 5
lim
= 3.
n→∞ 2n2 − 3n
√
n
= 0.
lim
n→∞ n + 1
n2
lim
= 0.
n→∞ n!
If |x| < 1, then limn→∞ xn = 0.
(a) lim
n→∞
(b)
(c)
(d)
(e)
(f)
4. Show that each of the following sequences is divergent.
(a) an = 2n.
(b) bn = (−1)n .
(c) dn = (−n)2 .
5. Suppose that lim sn = 0. If (tn ) is a bounded sequence, prove that lim(sn tn ) = 0.
6. Prove or give a counterexample: If (sn ) converges to s, then (|sn |) converges to |s|.
7. Suppose that (an ), (bn ), and (cn ) are sequences such that an ≤ bn ≤ cn for all n ∈ N and such that
lim an = lim cn = b. Prove that lim bn = b. (This is called the squeeze theorem.)
Limit Theorems
8. Suppose that lim an = a and lim bn = b. Let sn =
a3n + 4an
a3 + 4a
.
Prove
that
lim
s
=
carefully,
n
b2n + 1
b2 + 1
using the limit theorems.
1 − an+1
for a 6= 1.
1−a
(b) Find limn→∞ (1 + a + a2 + · · · + an ) for |a| < 1.
9. (a) Verify 1 + a + a2 + · · · + an =
(c) Calculate limn→∞ (1 +
1
3
+
1
9
2
+ ··· +
1
3n ).
n
(d) What is limn→∞ (1 + a + a + · · · + a ) for a ≥ 1?