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Now calculus rears its not-so-ugly head. Sequences are like bulls at a rodeo waiting to be lassoed, but the divergent ones can't be caught. Let's make sure we're comfortable with limits, and let's see which sequences we can stop.
In some of the sequences we graphed, it looked like as n got bigger the values an approached some particular value.
The terms of the sequence approach 0 as n approaches ∞.
When the terms of a sequence approach some finite value L as n gets bigger, we say the sequence converges to L, as is lasso. In symbols, a sequence converges to L if . This is just like convergence for functions. These are the bulls we can wrangle.
The sequence converges to 1, because
If we look at a convergent sequence on a number line, it looks like the dots are getting closer and closer to value L.
The sequence converges to 0. The dots are trying to get to 0 on the number line.
If we look at a convergent sequence on a 2-D graph, it looks like a function with a horizontal asymptote. The dots will get closer and closer to height L as n gets bigger.
The sequence converges to 1. As n gets bigger and we move to the right on the graph, the dots get closer and closer to height 1.
Just like a function, if a sequence doesn't converge, we say it diverges. Sequences can diverge for different reasons.
The sequence an = n diverges because as n approaches ∞ the terms an approach ∞ also. Since the terms aren't getting closer to anything finite, we say the sequence diverges. This is the bull that gets away from you before you can lasso it.
The sequence an = (-1)n diverges because it's indecisive and can't make up its mind whether to be + 1 or -1. This is the bull that catches you and throws you over it's head using its horns. You'll get up, try to wrangle it, and it'll just throw you over its head again.
To determine if a sequence converges or diverges, see if the limit
exists and is finite. For the sake of intuition, it may be helpful to graph the sequence.
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 monotonic because the terms are going only one way. We'll have a monotonically increasing California roll, extra wasabi on the side.
The sequence an = n is increasing because the terms get larger as n gets larger.
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.
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.
an ≤ M
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.
an ≥ K
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.
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 explain things out 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.
Any finite sequence is bounded.
Proof. A finite sequence has some largest term and some smallest term. These give upper and lower bounds, respectively.
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 ≤ an ≤ L + 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.
No, we aren't talking about dragons or dimes. We can use a scorpion for comparison, nasty little creatures that they are.
In the proof that a convergent sequence must be bounded, we said that "eventually" the terms had to be close to the limit of the sequence. The idea of "eventually" shows up a lot when talking about sequences and series.
We can split any sequence into two parts at any term am (assume M > 1). The head of the sequence consists of the finitely many terms before am. This is like the head of a scorpion. We care about where his head is at, but only so we can find his tail. The tail of the sequence consists of the infinitely many terms starting at am and continuing forever.
We really only care where the business end of a scorpion, the stinger on his tail, is at. Likewise, when talking about whether a sequence is bounded, we only care about its tail. There are only finitely many terms in the head of the sequence. Those finitely many terms must be bounded, because we can take the biggest and smallest terms to get upper and lower bounds. The only way the sequence can be unbounded is if the terms in the tail zoom off to ±∞.
Just like boundedness, when talking about whether a sequence converges, we only care about its tail. A sequence converges if its terms approach a finite limit L as n approaches ∞. The finitely many terms in the head of the sequence have no bearing on what happens to the terms an as n approaches ∞.
Tails will be very important when we get to series, so it's a good idea to start getting comfortable with them now.