Power Series — Taylor & Maclaurin
Representation of functions as infinite series and their applications.
Overview
A power series is an infinite series of the form $\sum_{n=0}^{\infty} c_n (x-a)^n$. Taylor and Maclaurin series allow us to represent smooth functions as power series, approximating them near a specific point using the function's derivatives.
Power Series
General Form: $$\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots$$
where $x$ is a variable, $c_n$ are coefficients, and $a$ is the center.
Radius of Convergence: The distance $R$ from the center $a$ within which the series converges. The series represents, approximates and defines functions converging within this radius.
Taylor Series
A Taylor series is an infinite series that represents a smooth function as a polynomial, approximating it near a specific point using the function's derivatives.
The Taylor series of $f(x)$ about center $x = a$: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$
$$= f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$
Conditions: $f$ must be infinitely differentiable at $a$.
Maclaurin Series
A Maclaurin series is a special case of the Taylor series in the region near $x = 0$. It provides polynomial approximations of functions like $\sin x$, $e^x$ and $\ln(1+x)$ using their derivatives at $x = 0$. Also called "Taylor at zero".
$$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots$$
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$
Constructing Series from Standard Forms
By applying algebra to standard Maclaurin or Taylor series, we can derive series for related functions:
- Substitution: replace $x$ with $x^2$, $-3x^2$, etc. (e.g. $e^{-3x^2}$)
- Multiplication: multiply two known series (e.g. $x^4 e^{-3x^2}$ or $(\sin 3x^2)e^{2x}$)
- Term-by-term integration: integrate a known series to approximate integrals like $\int \frac{\sin x}{x},dx$
Common Maclaurin Series
| Function | Series | Valid For |
|---|---|---|
| $e^x$ | $\sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ | all $x$ |
| $\sin x$ | $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ | all $x$ |
| $\cos x$ | $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$ | all $x$ |
| $\ln(1+x)$ | $\sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ | $-1 < x \leq 1$ |
| $\arctan x$ | $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$ | $-1 \leq x \leq 1$ |
| $(1+x)^n$ | $\sum_{r=0}^{\infty} \binom{n}{r} x^r$ | $ |
Applications
Function Approximation
- Taylor polynomials approximate functions near a point; the center $a$ is the anchor point of highest accuracy
- More terms give better approximations
Numerical Calculations
- Computing $e$, $\pi$, logarithms, trigonometric values
- Error estimation using remainder term
Calculus Operations
- Differentiating and integrating power series term by term
- Solving differential equations
- Evaluating limits
Error Estimation
Taylor's Remainder Theorem: $$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$$ for some $c$ between $a$ and $x$.