taylor series
TRANSCRIPT
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.
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
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.