HOMEWORK 1
A sequence (xn ) is bounded if there exists M > 0 such that |xn | ≤ M for all n ≥ 1.
(1) Let (xn ) be the sequence defined by
1
1
1
xn = 2 + 2 + · · · + 2 .
1
2
n
Show that (xn ) is convergent:
(a) Show that (xn ) is increasing.
(b) Show that (xn ) is bounded. (Hint: n2 > n(n − 1), for all n ≥ 2.)
(2) Let (xn ) be the sequence of real numbers defined by
√
√
xn+1 = 2 + xn , x1 = 2.
(a) Show that (xn ) is increasing. (Use induction).
(b) Show that xn < 2 for all n ≥ 1. (Use induction).
(3) Find the limit of (an )
1 − 5n4
(a) an = 4
.
n
+ 8n3
√
(b) an = n 2n + 3n .
(4) Find the sum of the following series:
∞
X
2n + 1
(a)
.
n2 (n + 1)2
n=1
∞
X
4
(b)
.
(4n − 3)(4n + 1)
n=1
(5) Suppose that x0 = 1 and x1 = 2. Define
xn−1 + xn−2
, n ≥ 2.
xn =
2
Compute lim xn
n→∞
(6) Let x0 = 1. Define
1
, n ≥ 1.
2 + xn−1
Suppose that we know (xn ) is convergent. Find lim xn .
n→∞
x
x
x
(7) Suppose xn = cos cos 2 · · · cos n . Find lim xn .
n→∞
2
2
2
(n+1)2
X 1
√ .
(8) Let xn =
k
2
k=n
(a) Show that
√
√
√
√
1
2 k+1−2 k < √ <2 k−2 k−1
k
for all k ≥ 1.
(b) Use (a) to compute lim xn .
xn = 1 +
n→∞
1