AcademySeries And Approximations
Academy
Taylor Series
Level 1 - Calculus topic page in Series And Approximations.
Principle
A Taylor series represents a function by matching all derivatives at a chosen centre. It builds a local polynomial approximation with infinitely many terms.
Notation
\(a\)
centre of the Taylor series
\(f^{(n)}(a)\)
nth derivative of f at a
\(n!\)
factorial of n
The Core Method
The Taylor series of \(f\) about \(a\) is
Taylor series
\[f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\]
When \(a=0\), this is called the Maclaurin series. A Taylor polynomial stops after finitely many terms.
Worked Cases
Question
Write the Maclaurin series for \(e^x\).
Answer
Every derivative of \(e^x\) is \(e^x\), so \(f^{(n)}(0)=1\) for every \(n\). Therefore \(e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots\).
Examples
Question
Find the degree 2 Maclaurin polynomial for \(\cos x\).
Answer
The Maclaurin series begins \(\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots\). Keeping terms through degree 2 gives \(1-x^2/2\).