Consider the sequences defined as follows: a_n = (-1)^n + 1, b_n = -1/n, c_n = 2
ID: 3110552 • Letter: C
Question
Consider the sequences defined as follows: a_n = (-1)^n + 1, b_n = -1/n, c_n = 2n, d_n = 3n + 1/4n - 1. (a) For each sequence, give an example of a monotone subsequence. (b) For each sequence, give its set of subsequential limits. Justify your answer. (c) For each sequence, give its lim inf and lim sup. Justify your answer. (d) Which of the sequences converges? Diverges to +infinity? Diverges to -infinity? (You do not need to justify your answer.) (e) Which of the sequences is bounded? (You do not need to justify your answer.)Explanation / Answer
a) For an we can have a monotone subsequence which has 1 as every term, i.e (an)k = 1.
For bn we can have entire sequence as it is increasing sequence.
For cn we can have entire sequence as it is increasing.
For dn we can have entire sequence as it is decreasing.
b) for an we have {-1, 1}. as constant sequences converge to one of these.
for bn it is {0} since the sequene is converging.
for cn it is {} as the sequence is diverging.
for dn it is {3/4} as the sequence is converging.
c) lim inf = -1, lim sup = 1 (for an)
lim inf = 0, lim sup = 0 (for bn)
lim inf = infinity, lim sup = infinity.
lim inf = 3/4 lim sup = 3/4
d) answered in part b
e) only cn is not bounded.