Let's use the Monotone Convergence Theorem.
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Well here's a complicated
sequence. It's a sequence that starts with its first
term equal to 1 and subsequent terms, a sub n plus 1, we define in terms
of a sub n. Will be the square root of a sub n plus 2.
This sequence is bounded above by 2. Well let's see why.
Let's suppose that a sub n is between 0 and 2.
Then what do I know about the next term, a sub n plus 1?
Well, then a sub n plus 1 must also be
non-negative, and that's just' cause of the square root of something.
But how big can a sub n plus 1 possibly be?
Well, a sub n plus 1 is the square root of a sub n, which could be as big as
2 plus 2.
So a sub n plus 1 can be as big as
the square root of 2 plus 2, which is equal to 2.
And this is great. Right?
This is telling me that if I know that a particular term is stuck
between 0 and 2, then the next term is also stuck between 0 and 2.
And what do I know about the first term?
Well, the first term is definitely between 0 and 2.
Right? A sub 1 is between 0 and 2.
And that means, that the next term must also be between 0 and 2.
And that means that the next term, a sub 3 must be between 0 and 3.
And that means that the next term, which is a sub 4 must be between 0 and 2.
That means that the next term,
which is a sub 5 must be between 0 and 2, and so on.
And what this actually is telling me, is that no matter
what n is, a sub n is between 0 and 2.
And the sequence is nondecreasing.
To show that this sequence is nondecreasing, I'm going to show that
a sub m is less than or equal to the next term,
a sub m plus 1.
And of course, I've got a formula for the next term.
That's just the square root of a sub m plus 2.
So the question, is this true?
Is a sub n less than or equal to the next term, the square root of a sub n plus 2.
And since a sub n is non-negative, I can
figure out when this happens, just by squaring both sides.
So, I'm wondering,
is a sub n squared, less than or equal to a sub n plus 2?
I'm going to subtract a sub n squared from both sides.
So I'm wondering, when is negative a sub n
squared, plus a sub n, plus 2 non negative.
Now, I can factor this.
This is asking when is 0 less than or equal to, how does this factor.
Got a 2 there, so I'll write a 2,1, 2 minus a sub n, 1
plus a sub n, is how this quadratic polynomial factors.
So the question is, when is this product non-negative?
Well in order for this product to be non-negative, either one
of these is 0, or they're both positive or they're both negative.
It can't happen, that both these
terms are negative.
So the only possibility is that a sub n is 2, or
a sub n is minus 1, or both of these terms are positive.
And that happens when a sub n is between minus 1 and 2.
So the upshot of this whole argument.
is that as long a sub n is between minus 1 and 2, then
a sub n is less than or equal to a sub n plus 1.
But I know that a sub n is between minus 1 and 2, because just
little while ago, we saw that a sub n was stuck between 0 and 2.
And consequently, this sequence is nondecreasing.
What does that all mean?
So get the sequence, and the terms of the sequence are between 0 and 2, and that
means that the sequence is bounded. And the sequence is nondecreasing.
Means the sequence is Monotone.
So, we've got a Monotone bounded sequence, and by the Monotone
Convergence Theorem, that means that the limit of the sequence exists.
We can try to get some numerical evidence to, guess the limit.
So, the first term in this sequence, was just defined to be 1.
To get the next term, I use this recursive definition.
I'll add 2 and take the square
root, and that gives me the a sub 2 term,
and that's the square root of 3, which is about 1.73.
Now to compute the next term, what do I do?
Well, using this formula, I'll add 2 and then take the square root of all
that, that will give me the a sub 3 term, and that's about 1.93.
Now to compute the
a sub 4 term. Right?
It's the same sort of process, I'll add 2.
And then take the square root of all that, and that'll give me the a sub 4 term.
And that's approximately 1.98. And if I keep on going, well
you might believe then, that the limit of a sub
n, as n approaches infinity, is in fact, 2.
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