And now we will do a very funny thing.

Here is this function:

Well, there is nothing funny so far.

Now let’s subtract from it

Next, add to it

If we keep subtracting and adding these weird things...

then step by step a familiar curve will take shape.

This curve is .

The explanation for the mysterious appearance of is:

By adding various powers of x, we can recreate a whole variety of functions.

Let's see how.

Let be differentiable k times on interval I that contains number a. Then, the Taylor polynomial of order k generated by function at point is:

Let's see an example.

Here is, for instance, function . Let’s find the Taylor polynomial of order 6 at point .

Ingredients:

The nth derivative at :

The zeroth derivative is the function itself.

We do not indicate the third derivative like this: , as we don’t use for the eighth derivative either. It’s easy to see why.

And now comes the Taylor polynomial:

The Taylor polynomial of function around zero

is useful, because it provides a good approximation of the function around zero.

For example, if we want to figure out the value of

without a calculator, then since near zero

Now let’s see,

The same thing using a calculator:

In fact, the calculator itself is using a Taylor polynomial to find the exact value of , except that it is not using a 4th-order polynomial, but of a higher order.

Next, calculate .

Well, we are not that lucky this time...

To be honest, the difference is a bit too much.

The truth is that a Taylor polynomial generated around provides a good approximation only for numbers close to zero. If a number is farther away from zero, we can do two things.

The first option is using a much higher order Taylor polynomial.

The more terms of the Taylor polynomial we generate, the longer section of the function takes shape.

The second option is to stick with the fourth-order version, but move it closer to 4.

We regenerate the Taylor polynomial, but this time around a number close to 4.

For instance, seems good.

It is a good idea to pick a number where and takes a value that is easy to compute.

Conveniently, is like that: and

Now we can get down to the Taylor polynomial.

If we want to compute the value of even more precisely, then we would have to generate more terms of the Taylor polynomial.

The more terms we generate, the more decimals will be accurate for .

If we are extremely greedy, we should generate all of them.

Well, that means an infinite number of terms, and what we get this way is called a Taylor series.

Let be differentiable any number of times on interval I that contains number a. Then, the Taylor series generated by function at point is:

Here is, for instance

Let's see the fifth-order Taylor polynomial and Taylor series at point .

The Taylor polynomial only approximates the original function;

the Taylor series, on the other hand, is identical to the function itself at every single point.

The Taylor polynomial generated by function at provides a good approximation of the function around number .

The more terms of the Taylor polynomial we generate, the longer section of the function takes shape.

If we get into generating the Taylor polynomial so much that we forget to stop...

well, then we get an infinitely large number of terms, and we called that Taylor series.

The Taylor polynomial only approximates the original function; the Taylor series, on the other hand, is identical to the function itself at every single point.

Let's see the Taylor series of a few functions.

Let’s start with , for instance.

Let’s generate its Taylor series at .

Taylor series where are often called Maclaurin series, too.

We can figure out the remaining terms based on this.

TAYLOR SERIES OF A FEW SPLENDID FUNCTIONS