Calculus is one of the grandest achievements of human thought, explaining everything from planetary orbits to the optimal size of a city to the periodicity of a heartbeat. This brisk course covers the core ideas of single-variable Calculus with emphases on conceptual understanding and applications. The course is ideal for students beginning in the engineering, physical, and social sciences. Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach. In this fifth part--part five of five--we cover a calculus for sequences, numerical methods, series and convergence tests, power and Taylor series, and conclude the course with a final exam. Learners in this course can earn a certificate in the series by signing up for Coursera's verified certificate program and passing the series' final exam.
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Calculus: Single Variable Part 5 - Discrete Calculus
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A Calculus for Sequences
It's time to redo calculus! Previously, all the calculus we have done is meant for functions with a continuous input and a continuous output. This time, we are going to retool calculus for functions with a <i>discrete</i> input. These are <i>sequences</i>, and they will occupy our attention for this last segment of the course. This first module will introduce the tools and terminologies for <b>discrete calculus</b>.
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Robert GhristProfessor
Mathematics and Electrical & Systems Engineering
Welcome to Calculus.
I'm Professor Ghrist.
We're about to begin Lecture 45 on sequences.
We now begin the last chapter of our course,
where we build a discrete calculus, reinventing everything that we've done so
far for functions with a digital input.
These functions, or sequences, will be familiar objects to you from throughout
the course, but we'll look at them from a new perspective of discrete calculus.
Everything that we have done thus far in this course has been for
functions that might be described as analog.
They are functions whose inputs and outputs, with few exceptions,
vary continuously, if not smoothly.
This allows us to talk about limits, differentials, and builds up calculus.
In contrast, so much of what we see and build is digital,
and not analog, built of discrete, or quantized bits.
Whether it's music, imagery, or information,
many things are not amenable to smooth calculus.
For this reason we're going to reconsider the notion of a sequence.
Now we have seen sequences before, but here is a slightly different perspective.
A sequence is a discrete or digital input function with an analog output.
That is, the inputs are natural numbers.
The outputs are reals.
Now sequences are everywhere, from economic data to digital signals.
But thinking of them as a function with an input, say n,
and an output, a sub n, is going to be the focus this chapter.
The notation that we'll use sometimes is writing out the terms of the sequence.
That is the outputs, a sub 0, a sub 1, a sub 2, etc.
Now our goal is going to be to redo everything that we have done
in this calculus class but in a digital form, or a discrete form,
for functions with a discrete input, that is for sequences.
All of the many things that we have discovered so far have
digital images in this discrete calculus.
We'll begin as we began this course,
with examples of interesting functions or sequences.
For example there are many sequences that are polynomial, like n squared.
Other sequences do not grow like a polynomial but
are rather exponential, like the sequence 2 to the n.
How would you describe the sequence 1, -1, 1, -1, repeating?
Well seems a little odd at first, but
one could describe this as a trigonometric function, for
example as cosine of n times pi.
Now more interesting sequences are out there.
For example, what if we remove the pi and consider the sequence cosine n?
If we plot the terms of that sequence as a function of n,
we see a curios mixture of wavelike behavior.
But non-repeatability the sequence never repeats and
in that way it's a little bit unlike the trigonometric function cosine.
This phenomenon is rather curious.
You've seen it before if you've ever seen an object that spins so
fast that your visual refresh rate can't keep up with it and
it looks like it's spinning at a different rate or even in a different direction.
This is sometimes called aliasing.
Let's begin our construction of discrete calculus with the notion of a limit.
Now what does it mean to take the limit of a sequence?
Well, I believe our adventure begins with failure because of the discrete input.
It doesn't make sense to talk about getting as close as you want to n = 11.
The one circumstance under which it does makes sense, is to take a limit
as n goes to infinity in this setting.
Then the original definition of the limit as n goes to infinity makes sense.
We say that that limit is L if for every epsilon
we find some value, let's say capital M,
past which the output of the function lies within epsilon of the limit.
This has to continue for any value of epsilon that is desired.
As we change the tolerance on the output we can find a new
tolerance on the input in order to satisfy the condition.
So with that in mind, why would be bother taking limits of sequences?
Well you've done so already.
In the context of Taylor series and approximation,
we began this very course with the discussion of what
e was in terms of the limit of a sequence.
e is 1 + 1 + one-half + one-sixth, and
each of these finite sums can be thought of as
a term in a sequence whose limit is e.
At other times in this course,
we've implicitly assumed that we can take a limit of a sequence.
We did so in Newton's method,
where we glibly stated that we hope the limit
of this sequence of points converges to the root that we were looking for.
Now all of this is good motivation, but how does one compute limits?
The principle that we'll see over and
over again is that one can use continuous methods to solve discrete problems.
Let's see a simple example.
Consider the sequence, a sub n = (1 + alpha over n) to the nth power.
You've seen this before in the context where n is x, a continuous variable.
Solving for the limit as n goes to infinity, follows the exact same method.
We take the log of both sides, pull that log inside the limit, and
use it to remove the exponent n out in front.
Then we use our knowledge of Taylor series to expand that logarithm
as alpha over n +big O of 1 over n squared.
And this is the key point, that our use of Taylor series and
big O works just as well in the discrete setting as in the continuous.
And so we can evaluate the limit, exponentiate, and
get an answer that should be very familiar.
Another example would be a sub n = 2 n squared-
n square root of quantity 4n squared + 5.
We would attack this by doing some algebraic
manipulation, factoring out a 2 n squared.
What is left is of the form 1- square root of 1 + 5 over 4n squared.
Here we see another Taylor Series lurking, that involving the square root.
Using the binomial series we can obtain,
after a little bit of algebraic simplification,
that the leading order term is negative five-fourths.
Everything else is in big O of (1 over n squared).
And so when we take the limit, we obtain negative five-fourths.
Now not every limit is so transparent.
For example, what would you do if asked to evaluate something of the form square root
of 2 + square root of 2 + square root of 2 + square root of 2, etc., all nested.
Well, the appropriate way to handle something like this is to realize it
as the limit of a sequence where we build things up step by step,
beginning with root 2 and then with root 2 + root 2 etc.
We want to compute the limit of this sequence, a sub n.
Now in order to do so, there's one difficult and critical step.
That is, you need to determine the recursion relation that the terms satisfy.
In this case, I'll tell you that it is the following.
a sub n = square root of 2 + a sub n-1.
Let's say I give you that recursion relation.
How do you compute the limit?
Let us, as before, denote this limit by L and
then apply the limit to the recursion relation.
On the left, we have the limit, as n goes to infinity, of a sub n.
That's L.
On the right we have an a sub n-1 term.
What's the limit of that as n goes to infinity?
Well of course, it too must be L.
Now we're assuming that we can slip that limit in under the square root,
let's forget about whether that's legal or not, and
see what happens when we try to simplify this algebraically.
Squaring both sides, we obtain L squared = 2 + L.
With a little bit of rearrangement, we get a polynomial
that easily factors into (L-2) and (L+1).
Now, this polynomial has two solutions,
namely negative 1 and positive 2.
We know that the limit of this sequence is not a negative number.
And therefore, if the limit exists, it must be equal to 2.
Indeed the limit does exist and is 2.
Let's look at another limit of this form.
In this case, 1+1 over 1+1 over 1+1,
etc., all nested together.
As before we can write out a sequence that gets closer and closer to this.
a naught is 1, a1 is 1 over 1 + 1, a2 is 1 over 1 +
1 over 1 over 1 or something like that, I don't know.
It keeps going.
The tricky part in this case is to find
the recurrence relation that these terms satisfy.
Once again, I’m going to tell you what it is.
It is that a sub n = 1 + 1 over a sub n-1.
This instruction tells you how to build the next term.
In order to compute the limit of the a sub ns, we follow the same
procedure as before, taking the limit of the recursion relation,
substituting in L for a sub n, and a sub n-1.
And then, with a little bit of algebraic rearrangement, what do we see?
We get L squared- L- 1 = 0.
This has roots 1 plus or minus square root of 5 over 2.
One of these is negative, the other positive.
We'll take that positive square root, and that is our limit.
This particular value is of some independent interest.
It is sometimes called the golden ratio and given the symbol phi.
It equals 1 + root 5 over 2.
Now perhaps you've seen the golden ratio before in
the context of some other sequences.
Perhaps you've seen the Fibonacci sequence,
that begins with 0, 1, 1, 2, 3, 5, 8,
13, 21, 34, 55, 89, etc.
There's a lot of interesting mathematics hiding behind here.
And if you'd like, you might want to take a look at the bonus material for
a hint at what calculus will be able to tell us about the Fibonacci sequence.
We began this course in calculus with a discussion of functions and
a contemplation of the exponential function.
In the same manner, we've begun our construction of the discrete calculus
by considering functions or sequences.
In our next lesson we'll look at the discrete analog of the exponential
function, and see how it relates to derivatives.
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