Taylor Series

#folder

Motivation

We can approximate non-polynomials using polynomials, and polynomials are much easier to work with.

Maclaurin's Approach

Example: Approximating cosine

Question

How would we find a quadratic polynomial in the form to approximate for small values of ?

left=-6; right=6;
top=2; bottom=-2;
---
y = \cos(x)
y = 1 - \frac{1}{2}x^2
Answer

We need to find the constants .
First, at since , then should also equal 1, so , so we have

Now, we want the polynomial to have the same tangent slope as .

left=-6; right=6;
top=2; bottom=-2;
---
y = \cos(x)
y = 1 - \frac{1}{2}x^2
y = 1 \{-\pi < x < \pi\}

Differentiating our equations:

so our derivative must be flat (0) at .

and since , , so we have .

This pattern continues, and we take the second derivative as well for the term:

and now we have to match the second derivative of the polynomial as well:

and since , , so we have

Remark

As you might imagine, as the degree of the polynomial increases, we have to higher order derivatives, and we can get a more accurate approximation for a larger domain.

Investigation

What if we try a cubic? Then we have:

so .

As for a quartic function,

and

so we have

which is an even better approximation:

left=-6; right=6;
top=2; bottom=-2;
---
y = \cos(x)
y = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4

Notice a few things:

Note

If we're going to approximate near for an example, we would write our polynomial in terms of :

Intuition

Recall using the linear approximation to approximate functions. We ask if we can do better; can we use a tangent polynomial of some higher degree to get an even better approximation? In fact, the tangent approximation is just a Taylor Polynomial with .

Maclaurin Polynomial or Taylor Polynomial at x = 0

Definition

To approximate a function at , the Taylor polynomial of the th degree looks like:

(see differentiation, factorial)

Or:

(see Summation and Product Notation)

Intuition

The constant term ensures the value of the polynomial matches the value of , the next term ensures the slope of the polynomial matches the slope of , and so on.

Taylor Polynomial

Definition

To approximate a function at , the Taylor polynomial of the th degree looks like:

Or:

Remark

"Maclaurin Polynomial" or "Maclaurin Series" is just a Taylor Polynomial/Series "centred" at . The "centre" of the series is where we take our derivatives from.

So a Maclaurin series is a special case of a Taylor series, and a Taylor series is a generalization of a Maclaurin series. You may see "Maclaurin series" used quite often from other people, but I will just refer to them as "Taylor polynomials at ".

Taylor Series and Maclaurin Series

Definition

What if we kept adding terms infinitely? Then we have an infinite series, and we would get an ever more accurate approximation. That is, a Taylor/Maclaurin series is simply a Taylor/Maclaurin polynomial with .

Convergence and Divergence

Info

We therefore can say that the sequence converges to some value. Then you would say that the infinite sum is equal to the value that it's converging to.

However, sometimes we can choose a point, and no matter how many derivatives we use, we will never converge to that value. If this occurs, we say the series diverges.

When a Taylor series is convergent, then the Taylor series is equal to the function it is derived from.

More details at Convergence and Divergence.

proof
None, it is impossible. In fact, this isn't necessarily true. Consider this piecewise function from a famous example:

Using the definition of the derivative, we see that for all , , so the MacLaurin series would be 0. This obviously converges for all , yet does not match for any .

Well, that's disturbing.
- Prof. David Harmsworth

However, in practice, we are very unlikely to encounter this.

Uniqueness

Theorem

Taylor Series are unique for each , , and .

Furthermore, if a polynomial matches the values of and its derivatives at a point, then that polynomial must be the Taylor polynomial.

Examples

Example - Natural Number

Find the Taylor Polynomial of at

solution

Strange property: if we infinitely add, we converge to a number.

top=30; bottom=-2;
left=-15; right=10;
---
y = e^x
y = 1 + x + x^2/2! + x^3/3! + x^4/4!

Example - Sine

Find the Taylor Polynomial of at

solution
Notice:

Which gives:

top=2; bottom=-2;
left=-10; right = 10;
---
y = \sin(x)
y = x - x^3/3! + x^5/5! - x^7/7!
Example - Cosine

Find the Taylor Polynomial of at

solution
Listing out the derivatives again:

Which gives:

top=2; bottom=-2;
left=-10; right = 10;
---
y = \cos(x)
y = 1 - x^2/2! + x^4/4! - x^6/6!
Example - Euler's Formula

Euler and complex numbers:

Euler noticed that this follows the same pattern as the derivatives of sine/cosine (i.e as you take more derivatives, it "cycles" through).

which is Euler's Formula!

Note that we got the values for and from the two examples above.

Example - Natural logarithm

Find the Taylor Polynomial of

where .

solution

In the Taylor polynomial:

Remark

Using this with , if we try , we have on the left ride, and if we sub in the series we just got above, into the series equations for (#^ee7acd), everything cancels to as well on the right.

left=-5; right=10;
bottom=-5; top=5;
---
y = \ln(1 + x)
y = x - x^2/2 + x^3/3 + x^4/4 + x^5/5 | #388c46
y = x - x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 + x^7/7 + x^8/8 + x^9/9 + x^{10}/{10} | #6042a6
y = x - x^2/2 + x^3/3 + x^4/4 + x^5/5 + x^6/6 + x^7/7 + x^8/8 + x^9/9 + x^{10}/{10} + x^{11}/{11} + x^{12}/{12} + x^{13}/{13} + x^{14}/{14} + x^{15}/{15} + x^{16}/{16} + x^{17}/{17} + x^{18}/{18} + x^{19}/{19} + x^{20}/{20} | #fa7e19
(4, -2) | label: 1 - x^2/2 + ... + x^5/5 | #388c46 | cross
(4, -3) | label: 1 - x^2/2 + ... + x^{10}/{10} | #6042a6 | cross
(4, -4) | label: 1 - x^2/2 + ... + x^{20}/{20}| #fa7e19 | cross

This Taylor series diverges, and does not approach as

Historically, this is is what Newton did before Taylor, using the function:

$

Newton's binomial theorem

Newton's Binomial Theorem

Newton generalized the binomial theorem for real exponents other than non-negative integers (the same also applies to complex exponents).

For this generalization, the finite sum is replaced with an infinite series.

For example:

He noticed that we could generate these expansions with this rule:


where is defined by the recurrence relation:

notice that for , all terms of .

Question

What if is not a positive integer? In that case, the series never terminates.

Example

We with to find the binomial expansion of

solution

Note that we have because our equation is actually .

As a matter of fact, this is exactly what we would get with a Taylor polynomial, centred at :

(see derivative)


However, Newton did now have this tool at his disposal.

How would he check that his result was correct? Newton multiplied both sides by themselves (i.e squared both sides).


Newton did this up to and found that past the first 2 terms, the rest cancelled out. He then assumed, without proof, that the rest of the terms would also cancel. The proof came later.

As such, this generalization became known as Newton's Binomial Theorem.

Note

This is the same as the standard binomial theorem, but generalized to allow .

Examples

Newton specifically wrote the following instances:

Example - Geometric Series

That is, we can add up an infinite number of numbers, and we will end with a number.

Which is Bananas
- Prof. Scott

Note this only works for .

E.g :

Which comes from Infinite Series#Geometric series

Important Taylor Series to Remember

All of these have been derived in the #Examples section

Important

Geometric Series

for

Natural Number e

for

Natural Log (1 - x)
Integral of geometric series

for

Cosine

for

Sine
Cosine with shift

for

Binomial Theorem

for

Two-Variable Taylor Series

Consider a function . For single variable functions, we would have taken derivatives.

For two variables, we will instead freeze one variable (say ), and becomes a function of . Then our Taylor series centred at is the same as the single-variable case, except we take a partial derivative instead:

Now, the terms , , , etc. aren't really constant. But we can expand each of them as a Taylor series centred at :

If we substitute each of these into (1), we get an infinite series of an infinite series. But we can re-organize these terms so it's just one series:

The first two lines are actually the tangent plane approximation formula.

After some factoring and adding a more terms, we arrive at the formula for a two-variable Taylor polynomial:

Definition

To approximate a function at , the Taylor polynomial of the th degree looks like:

where and .

These terms all show a pattern: the structure of a single-variable Taylor series is retained, but all partial derivatives must appear.

Moreover, the coefficients come from Pascal's triangle.
Why? Because the term actually represents . Recall Clairaut's theorem. Since continuity is a requirement for a Taylor series, we can say that . Similarly, is made up of , , and .