Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Numerical Methods

Lecture 1 – Taylor Series
Notes

Lecturer: Stephan Juricke
Date of lecture: February 3rd, 2022
Author: Lirik Maxhuni

Taylor Series Expansion
First and foremost, Taylor Series is a Power Series!
So, let’s start with a reminder of what a Power Series is:
A power series has a radius/interval of convergence
...

Example: Taylor Series for 𝑓 (𝑘)(𝑥) = 𝑒 𝑥 , so 𝑓 (𝑘) (𝑐) = 𝑒 0 = 1
...
e
...
:
𝑒𝑥 ≈

1 1
1
1
+ 𝑥 + 𝑥2 + ⋯ = 1 + 𝑥 + 𝑥2
0! 1!
2!
2

Taylor Series of a polynomial is the polynomial itself
...
, 𝑓 𝑖𝑠 (𝑛 + 1)𝑡𝑖𝑚𝑒𝑠 continuously differentiable over
[a, b]
...
e
...


For the special case where 𝑛 = 0, we have
𝑓(𝑥 ) = 𝑓(𝑐 ) + 𝑓′(𝜓𝑥 )(𝑥 − 𝑐 )
𝑏 = 𝑥, 𝑐 = 𝑎
𝑓 (𝑏) = 𝑓 (𝑎) + 𝑓′(𝜓𝑥 )(𝑏 − 𝑎)
𝑓′(𝜓𝑥 ) =

𝑓 (𝑏) − 𝑓(𝑎)
𝑏−𝑎

This result is known as Mean Value Theorem
...
e
...

Ex
...

Let’s prove that:
For any 𝑥 ∈ 𝑅 we can find 𝑠 ∈ 𝑅0 + so that |𝑥| ≤ 𝑠, and |𝜓𝑥 | ≤ 𝑠, because 𝜓𝑥 is
located between c and x
...

Ex
...


Another example
Compute cos(0
...
1, 𝜓𝑥 = ?
)𝑛+1

|(−1

𝑥 2(𝑛+1)
(0

---