Find an example of a sequence with the given property for each of the following:?

Question

a) The sequence is monotone but not bounded?
b) The sequence is bounded but not monotone?
c) The sequence is bounded and strictly increasing?
d) The sequence is convergent but not monotone?
e) The sequence is strictly decreasing but not convergent?
f) The sequence is neither bounded nor monotone?
g) The sequence is bounded below, not bounded above, and contains an infinite number of negative terms.

Answer 1

a) xn = n
b) xn = (-1)^n
c) xn = 1 - 1/n
d) xn = (-1)^n/n
e) xn = -n
f) xn = n*(-1)^n
g) xn = -1/n if n is odd, and n if n is even

Answer 2

a) x_n = n

b) x_n - sin(n)

c) x_n = 1 - 1/n

d) sin(n)/n (converges to 0)

e) x_n = -n

f) x_n = n sin(n)

g) x_n = -1/n, if n is odd, x_n = n if n is even