Taylor Series and Analytic Functions

Polynomials are the simplest functions to calculate, differentiate, and integrate. Taylor series provide a way to represent more complex, transcendental functions as “infinite polynomials,” allowing us to study them using the tools of power series.

Power Series

A Power Series centered at $a$ is an infinite series of the form: ${\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n=c_0+c_1(x-a)+c_2(x-a{)}^2+\dots$ For any such series, there exists a Radius of Convergence $R\in [0,\infty ]$ such that the series converges absolutely if $|x-a|<R$ and diverges if $|x-a|>R$.

• At the boundaries $|x-a|=R$, the series may or may not converge (requiring specific tests like the Ratio or Root tests).

Taylor and Maclaurin Series

If a function $f(x)$ is infinitely differentiable at $a$, we can define its Taylor Series as: $T(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$ When $a=0$, the series is specifically called a Maclaurin Series.

Common Maclaurin Series:

• $e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$ for all $x\in \mathbb{R}$.
• $\sin (x)={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$ for all $x\in \mathbb{R}$.
• $\cos (x)={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{(2n)!}$ for all $x\in \mathbb{R}$.
• $\frac{1}{1-x}={\sum }_{n=0}^{\infty }{x}^n$ for $|x|<1$.

Taylor’s Theorem and the Remainder

A crucial question in analysis is: Does the Taylor series actually equal the function? We define the $n$-th degree Taylor Polynomial $P_n(x)$ and the Remainder $R_n(x)$: $f(x)=P_n(x)+R_n(x)$ By Taylor’s Theorem (with Lagrange Remainder), there exists some $c$ between $a$ and $x$ such that: $R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a{)}^{n+1}$ $f(x)$ is equal to its Taylor series if and only if ${\lim }_{n\to \infty }{R}_n(x)=0$.

Analytic Functions

A function $f(x)$ is analytic at $a$ if it can be represented by a power series in some neighborhood of $a$.

• Not all smooth ($C^{\infty }$) functions are analytic! A famous counter-example is $f(x)=e^{-1/x^2}$ for $x\ne 0$ and $f(0)=0$. All its derivatives at $x=0$ are zero, so its Taylor series is identically zero, yet the function itself is non-zero for all $x\ne 0$.

Properties of Power Series

Within their radius of convergence, power series behave almost exactly like polynomials:

1. Term-by-term Differentiation: The derivative of a power series is the sum of the derivatives of its terms, and the radius of convergence remains the same.
2. Term-by-term Integration: The integral of a power series can be computed by integrating each term individually.
3. Uniqueness: If two power series represent the same function in a neighborhood of $a$, their coefficients must be identical.

Python: Approximating Functions with Taylor Polynomials

We can use Python to visualize how the Taylor polynomial $P_n$ approaches $f(x)$ as $n$ increases.

import math

def taylor_exp(x, n):
    """Computes the n-th degree Taylor polynomial of e^x at 0."""
    sum = 0
    for i in range(n + 1):
        sum += (x ** i) / math.factorial(i)
    return sum

x_val = 1.0
actual = math.exp(x_val)

for n in [1, 3, 5, 10]:
    approx = taylor_exp(x_val, n)
    print(f"n={n}: Approx={approx:.6f}, Error={abs(actual - approx):.6e}")

Significance

Taylor series are the backbone of numerical analysis and physics. They allow us to approximate complex potential fields, solve differential equations by assuming power series solutions (the Frobenius method), and define functions in the complex plane. They transform the infinitesimal study of derivatives into the discrete study of coefficients.

Exercise