Taylor Series

Taylor SeriesMaddie & Iggy

What is a Series?

Infinite Series

Geometric Series

Converging/Diverging

Power Series: We will be find a general method for writing a power series representation for a function.

Let’s Do Some Math

Construct a polynomial P(x) where:

P(0)=1

P’(0)=2

P”(0)=3

P’’’(0)=4

P’’’’(0)=5

So What?

Using polynomials makes calculus a lot easier.

An Example

Construct a polynomial P(x) to the fourth term that matches the behavior of ln(1+x) at x=0.

Undo the problem: Does it make sense?

This is a Taylor Polynomial

One More Example & Equation

Y=sin(x) Construct a 7th order polynomial.

(Pn(0)/n!) *xn

Write This Down

Maclaurin Series

Taylor Series

Real World (kinda)

Engineers will know the complicated function and they can break that down into polynomials when doing things like building bridges.

It’s *abstract*

Trying to Make Sense of This

We had a power series, we wanted to take the power series and be able to apply it to a function so it could represent a function, either at x=a or x=0.

When given the derivatives at P(0), we could solve for a polynomial, this helped us learn how to use the derivative values to build polynomials.

Then we looked at real functions, where we know the derivative and can make the same list like we had the first time. We constructed a polynomial that would adhere to this list.

We then had a polynomial that was a representation of a function using power series.

Euclid alone has looked on Beauty bare.

Let all who prate of Beauty hold their peace,

And lay them prone upon the earth and cease

To ponder on themselves, the while they stare

At nothing, intricately drawn nowhere

In shapes of shifting lineage; let geese

Gabble and hiss, but heroes seek release

From dusty bondage into luminous air.

O blinding hour, O holy, terrible day,

When first the shaft into his vision shone

Of light anatomized! Euclid alone

Has looked on Beauty bare. Fortunate they

Who, though once only and then but far away,

Have heard her massive sandal set on stone.

Work Cited

"Infinite Series." Calculus Ap Edition. Pearson College Div, 2013. 487-489. Print.