Assignment 5, Math 220
Due: Friday, March 8th, 2002
1 Use the − N definition of convergence to show that the sequence
an = 3n+1
converges.
2n+5
2 Suppose that (xn ) is a sequence of real numbers converging to x,
√
with xn ≥ 0 for all n ∈ N.√Show that the sequence ( xn ) of positive
square roots converges to x.
√
√
3 Let x1 = 2 and xn+1 = 2 + xn for n ∈ N. Show that (xn )
converges and find the limit.
4 Establish the convergence or divergence of the sequence yn , where
1
1
1
+
+···+
for n ∈ N.
yn =
n+1 n+2
2n
5 Give an example of an unbounded sequence with a convergent subsequence.
√
6 If xn = n, show that (xn ) satisfies lim |xn+1 − xn | = 0 but that it
is not a Cauchy sequence.
7 If x1 < x2 are arbitrary real numbers and xn = 12 (xn−2 + xn−1 ) for
n > 2, show that (xn ) is convergent. What is its limit?
8 By using partial fractions, show that
∞
X
1
1
= .
n(n + 1)(n + 2)
4
n=1
1