Big O Notation

A similar concept to Big O, but for instead of , we deal with .

It's a method/definition for qualitatively comparing the behaviour of two functions in a limiting regime. (not quantitate)

Definition

We say two functions and are related to one another via the order symbol (big O) if:

then as .

(see limit)

Example

Since we know that for all , we can state that:

Since it is also true that on the same interval, we can state that:

We can also state:

However, we CANNOT state:

but we can state:

Remark

Big O notation NEEDS the qualifier

  • As , higher exponents approach 0 more quickly, so our upper bound is the lower exponent, and the higher the "better" (i.e we don't always want to just pick )
  • As , higher exponents approach more quickly, so our upper bound is the higher exponent, and the lower the "better"
Example

Let . Then we can state:

Example

Since , we can state:

Example

Consider the rational function on the interval .

To find an upper bound, rearrange:

therefore:

In Conjunction with Taylor's Inequality

This is where things get crazy

Recall the definition of a Taylor polynomial:

Recall Taylor's inequality:

for some for all .

Using Big O notation we can easily write the Taylor remainder theorem:

Definition

We can rearrange:

Why can we do this?

Example

Remember, we seek the highest allowable exponent

Example

Example

Arithmetic with Taylor Polynomials

We can take repeated derivatives or multiply two series.

Example

Find the Taylor polynomial of .

solution

It's convenient to display this as a table:

So as .

Example

Find the Taylor series of up to as

solution

conclusion: at

Example

These types of Taylor polynomials are very useful for numerical estimation.

What is at ?

solution
Taylor series:

(see log laws)

Recall:

One option is to plug this in immediately:

Alternatively, we can convert these to an integral:

this is a geometric series, and we know

using the Taylor polynomial for tangent that we did before:

Proving a point: if we try . If we try in Mathematica, we get which is wrong.

Evaluating Limits

Utility comes whenever a qualitative estimate of the error is useful:

  1. Taking limits
  2. Manipulating Power Series
Example

Evaluate .

solution

Example

Evaluate .

solution

But is this legit?

Writing this with big O:

So it does work. We also could've re-written the limit as:

Example

Estimate at .

solution
We will expand as a Taylor series about .

Example

Evaluate .

solution
Note that at , looks like :

left=-5; right=5;
bottom=-1.5; top=1.5;
---
y = \sin(x - \pi)

Substitute with .

Example

Evaluate

solution
Notice:

Example

Special relativity gives mass as:

Rest energy:

Kinetic energy:

If , does the following hold?

Look at the Taylor polynomial as (since ) .

Which is the desired result.