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Introduction to Sequences - At A Glance:

Imagine a Kung Fu black belt took a function and chopped through it, leaving only discrete values. Those discrete values would form a sequence. Because the sequence is just a coarse-chopped list of numbers made from a function, the sequence acts in ways similar to functions.

Let's look at some of the ways the martial arts master can serve up a sequence to us. These meals are going to look similar to functions, so you may want to review it to see the similarities.

We say a sequence is increasing if the terms get larger as n gets larger. This is like a lop-sided sushi roll where the piece on the right is bigger than the one to its left.

In symbols, if m < n then am < an.

We say a sequence is decreasing if the terms get smaller as n gets larger. This is like a lop-sided sushi roll where the piece on the right is smaller than the one to its left.

In symbols, if m < n the am > an.

If a sequence is increasing or decreasing we call it \textbf{monotonic} because the terms are going only one way. I'll have a monotonically increasing California roll, extra wasabi on the side.

Sample Problem

The sequence an = n is increasing because the terms get larger as n gets larger.

Sample Problem

The sequence  is decreasing because the terms get smaller as n gets larger.

Be Careful: increasing and decreasing aren't opposites. It's possible for a sequence to be neither increasing nor decreasing.

Sample Problem

The terms of the sequence an = (-1)n bounce back and forth between 1 and -1. This sequence is neither increasing nor decreasing.

Be Careful: Using the word "increasing" to refer to a function is ambiguous because it could mean either nondecreasing or strictly increasing. We don't usually care about nondecreasing sequences. They are about as interesting as watching water evaporate off a hot road surface in the middle of summer. That's why, for sequences, we use "increasing" as an abbreviation for "strictly increasing".

Just like a function, we say a sequence is bounded above if all terms of the sequence are less than or equal to some value M.

In symbols,

anM

for all n.

No surprises here. We say a sequence is bounded below if there's a value K such that all terms of the sequence are at least K.

In symbols,

anK

for all n.

If a sequence is bounded above and below, we say it's bounded. For our sushi sequence, if it is bounded, we can make a bento box with it.

If a sequence is missing one or both of these bounds, then it's unbounded.

There are a couple of theorems connecting the ideas of boundedness and convergence for sequences. These are some of the ideas that spice up our sushi roll sequences.

Theorem

A monotonic bounded sequence must converge. This is a pretty obvious statement, so we could call this the california roll theorem. Everyone knows what it is, and as boring as it may be, everyone eats it.

Proof. Rather than be overly mathematical, we'll just draw pictures to give you the idea. If a sequence is increasing, the terms are going up.

If the sequence is bounded, the terms can't go up forever, because they can't go above the upper bound.

That means the sequence converges. If the upper bound given wasn't the best upper bound possible, the sequence could converge to some value L smaller than the given upper bound.

Similarly, if the sequence is decreasing the terms are going down. If the sequence is bounded, the terms can't go down forever, so they must approach some floor. That means the sequence converges.

Theorem

Any finite sequence is bounded.

Proof. A finite sequence has some largest term and some smallest term. These give upper and lower bounds, respectively.

Theorem

A convergent sequence must be bounded. We could call this the bento box theorem. If the roll converges to some size, it will fit in a box.

Proof. If a sequence converges to some value L, then eventually all the terms must be very close to L. In particular, eventually the terms must be within 1 of L in either direction.

Formally, when n gets large enough we have

L – 1 ≤ anL + 1.

There can be only finitely many terms, all at the beginning of the sequence, that aren't within 1 of L.

Take the largest term that isn't within 1 of L. If this term is bigger than L + 1, this term is an upper bound for the sequence.

Otherwise, L + 1 is an upper bound for the sequence.

Similarly, take the smallest term that isn't within 1 of L. If this term is smaller than L – 1, this term is a lower bound for the sequence.

Otherwise, L – 1 is a lower bound for the sequence.

In contrast, a bounded sequence does NOT have to converge. This non-theorem is like a dragon roll. It will always keep you on your toes. One of the easiest examples is the sequence

an = (-1)n.

This sequence is definitely bounded, since -1 ≤ an ≤ 1 for all n. However, the sequence can't converge because it's indecisive.

Example 1

Is the sequence  bounded or unbounded?


Example 2

Is the sequence an = -n2 bounded or unbounded?


Example 3

Is the sequence an = (-1)n n bounded or unbounded?


Exercise 1

Determine if the sequence is increasing, decreasing, or neither.

an = 2n

Exercise 2

Determine if the sequence is increasing, decreasing, or neither.

Exercise 3

Determine if the sequence is increasing, decreasing, or neither.

an = (-1)n n2

Exercise 4

Determine if the sequence is increasing, decreasing, or neither.

an = -n2

Exercise 5

Determine if the sequence is increasing, decreasing, or neither.

Exercise 6

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

Exercise 7

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

an = 4 – n

Exercise 8

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

an = (-1)n

Exercise 9

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

an = n3

Exercise 10

Determine if the sequence is (a) bounded above, (b) bounded below, and (c) bounded or unbounded.

an = (-1)n2n

Exercise 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.

Exercise 12

Determine if the statement is true or false. Explain your reasoning.

The sequence 1,1,1,1,... is both convergent and bounded.

Exercise 13

Determine if the statement is true or false. Explain your reasoning.

If a sequence diverges, the sequence is unbounded.

Exercise 14

Determine if the statement is true or false. Explain your reasoning.

If a sequence is unbounded, that sequence diverges.

Exercise 15

Determine if the statement is true or false. Explain your reasoning.

If a sequence converges, there is some value K such that Kan for all n.

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