This document provides an overview of Taylor series and power series. It discusses what a series is, examples of geometric and infinite series, and how to construct Taylor polynomials that match the behavior of functions like ln(1+x) and sin(x) at specific points by using their derivatives. The document notes that power series representations allow complicated functions to be broken down into polynomials, which can be useful for applications like engineering and building bridges.

1. Taylor Series
Maddie & Iggy

2. 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.

4. 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

5. So What?
Using polynomials makes calculus a lot easier.

6. 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

8. One More Example & Equation
Y=sin(x) Construct a 7th order polynomial.
(Pn(0)/n!) *xn

10. Write This Down
Maclaurin Series
Taylor Series

15. 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*

16. 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.

17. 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.

18. Work Cited
"Infinite Series." CalculusAp Edition. Pearson College
Div, 2013. 487-489. Print.