Determine if the sequence is increasing, decreasing, or neither.
an = 2n
Answer
As n gets larger the terms get larger, so this sequence is increasing.
Example 2
Determine if the sequence is increasing, decreasing, or neither.
Answer
As n gets larger the terms get smaller, so this sequence is decreasing.
Example 3
Determine if the sequence is increasing, decreasing, or neither.
an = (-1)n n2
Answer
The terms of this sequence bounce back and forth between positive and negative, so this sequence is neither increasing nor decreasing.
Example 4
Determine if the sequence is increasing, decreasing, or neither.
an = -n2
Answer
This sequence is decreasing, since it consists of negative numbers that are getting further and further from zero.
Example 5
Determine if the sequence is increasing, decreasing, or neither.
Answer
Since the fraction is surrounded by absolute value signs, the factor (-1)n has no effect on the term. We can rewrite the term
Since 2n grows more quickly than n, the terms are positive and approach zero. This is a decreasing sequence.
Example 6
Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.
Answer
Since , each term an is between 4 and 5.
a) Yes, this sequence is bounded above.
b) Yes, this sequence is bounded below.
c) This sequence is bounded.
Example 7
Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.
an = 4 – n
Answer
Each term is 4 minus a natural number, so all the terms are smaller than 3. The value of 4 – n can get as negative as we like.
a) Yes, this sequence is bounded above.
b) No, this sequence is not bounded below.
c) This sequence is unbounded because it has no lower bound.
Example 8
Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.
an = (-1)n
Answer
Each term of this sequence is between -1 and 1.
a) Yes, this sequence is bounded above.
b) Yes, this sequence is bounded below.
c) This sequence is bounded.
Example 9
Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.
an = n3
Answer
a) No, this sequence is not bounded above. The value of n3 can get as large as we like.
b) Since all terms are positive, this sequence is bounded below by 0.
c) This sequence is unbounded because it has no upper bound.
Example 10
Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.
an = (-1)n2n
Answer
The terms of this sequence alternate between positive and negative, getting farther from 0 with no bounds anywhere in sight.
a) No, this sequence is not bounded above.
b) No, this sequence is not bounded below.
c) This sequence is unbounded. It has no upper or lower bound.
Example 11
Determine if the statement is true or false. Explain your reasoning.
If a sequence has 5 ≤ an ≤ 6 for all n, then the sequence must converge.
Answer
False. As an example, all terms of the sequence
5,6,5,6,5,6,...
fall between 5 and 6. However, this sequence is indecisive so it doesn't converge.
Example 12
Determine if the statement is true or false. Explain your reasoning.
The sequence 1,1,1,1,... is both convergent and bounded.
Answer
True. This sequence converges to 1 and is bounded both above and below by 1. If you don't like the upper and lower bounds being the same, we can also say it's bounded below by 0.9 and above by 1.1 or something like that.
Example 13
Determine if the statement is true or false. Explain your reasoning.
If a sequence diverges, the sequence is unbounded.
Answer
False. To say a sequence diverges means it doesn't converge. The sequence
5,6,5,6,5,6,...
diverges because it's indecisive. However, this sequence is bounded (below by 5, above by 6).
Example 14
Determine if the statement is true or false. Explain your reasoning.
If a sequence is unbounded, that sequence diverges.
Answer
True. If a sequence is unbounded, the terms are going off to ∞ and/or -∞. Since the terms aren't approaching anything finite, the sequence must diverge.
Example 15
Determine if the statement is true or false. Explain your reasoning.
If a sequence converges, there is some value K such that K ≤ an for all n.
Answer
True. This statement is saying "if a sequence converges, it has a lower bound." This is true, because a convergent sequence must have both upper and lower bounds.
, each term an is between 4 and 5.