Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

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From the course by The Ohio State University

Calculus Two: Sequences and Series

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Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"

From the lesson

Taylor Series

In this last module, we introduce Taylor series. Instead of starting with a power series and finding a nice description of the function it represents, we will start with a function, and try to find a power series for it. There is no guarantee of success! But incredibly, many of our favorite functions will have power series representations. Sometimes dreams come true. Like many dreams, much will be left unsaid. I hope this brief introduction to Taylor series whets your appetite to learn more calculus.

- Jim Fowler, PhDProfessor

Mathematics

Holograms.

[SOUND].

They tell me that if you take a total little bit of a

hologram and you look through it, you can still see the whole picture.

The same is sort of true of analytic functions.

All right? Let's think about sine.

So here, I've got the graph of a function, y equals f of x.

And in this case, the function that I'm graphing is just sine.

I'm going to think about this graph a little bit

and just focus on this region right around the origin.

All right, imagine, if you will, that I've taken.

Away all of the graph and I'm just looking

at just this little tiny piece right around the origin.

What can I figure out about sine just by looking right around the origin.

Well, if I look around the origin, I can

figure out the value of sine at the origin.

If I'm just looking the origin I can

figure out the derivative of sine at the origin.

I can figure out the second derivative sine at

the origin.

I can figure out the third derivative of sine.

Just by looking at this little tiny piece of the graph.

I can figure out all of the derivatives of sine, just

by looking at this little tiny piece of the graph of sine.

And if I can calculate all the derivatives of sine, that means that just by looking

at that piece, I can write down the Taylor series for sine at 0.

And we've already seen that the Taylor

series for sine converges to sine everywhere.

That's exactly right.

We've seen that sine of x is equal to its Taylor series for all values of x.

And what that means is that just this little tiny piece of

sine, just around the origin, manages to encode the entire story about sine.

All of the values of sine can be recovered just

by knowing a little tiny piece of sine around the origin.

Because that's enough to calculate all the derivatives at 0 which

is enough to recover the Taylor series, which recovers the entire function.

Contrast that with something involving absolute value.

Well, here's the graph. Of a function, so y=f(x).

But in this case, the function is f(x)=|x-1|-1.

Now, what happens if I zoom in on the origin?

Well, this little piece of the graph just around the origin is enough to

calculate some facts about f, right?

What can I calculate just by looking right around the origin?

Well I can calculate the value of this function is 0 when I plug in

0, I can calculate that the derivative of this function is negative 1 at 0.

And I can calculate that all of the higher derivatives, all of the

second, third, fourth and so on derivatives at 0 are equal to 0.

So yeah, this thing looks like a straight line around here and

that infinitesimal information at the point 0.

Just the derivatives at the point 0 are enough to

recover the function around 0 on a whole interval around 0.

Now of course, I might have been tricked into

thinking that the entire function just looked like this, right.

I can't tell that there's this point where the function

fails to be differentiable, just when I'm looking over here.

But nevertheless, just this infinitesimal information

at a point, is still enough to recover some information

about the function in a whole interval, around that point.

Real analytic functions are really quite surprising.

Infinitesimal information, just at a point, is

giving you information at a whole interval.

Sine, cosine, and e to the x are even better.

In that case, the Taylor series converges to those functions everywhere.

Which means infinitesimal information at a single

point, on say the graph of sine, alright, a single point on the graph

of cosine, or at a single point on the graph of e to the x.

That information is telling you everything about the entire function.

Yeah en, en, entire is actually the word that, that's used.

Entire describes functions like sin where just the infinitesimal information at a

point is enough to recover a power series with an infinite radius of convergence.

Converges everywhere to the entire function.

So, it's really like a hologram.

Alright?

Just a little tiny bit of a hologram records everything

in the scene and so too would say sine of x.

Alright?

Just a little tiny piece of sine reveals everything about the function.

[SOUND]

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