6.3 Taylor and Maclaurin Series

Taylor and Maclaurin Series

Taylor and Maclaurin series let you represent complex functions as infinite polynomial expressions centered at a specific point. This is incredibly useful because polynomials are easy to differentiate, integrate, and evaluate, while the original functions (like $e^x$, $\sin(x)$, or $\ln(1+x)$) can be much harder to work with directly.

The core idea: if you know all the derivatives of a function at a single point, you can reconstruct the entire function (within some radius) as a power series. The more terms you include, the better your approximation gets.

Construction of Taylor Polynomials

A Taylor polynomial approximates a function $f(x)$ near a specific point $a$ by matching the function's value and its derivatives at that point. The $n$th-degree Taylor polynomial $P_n(x)$ is:

$P_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n$

Written in summation notation: $P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k$

A Maclaurin polynomial is just a Taylor polynomial centered at $a = 0$:

$P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots + \frac{f^{(n)}(0)}{n!}x^n$

Notice each term has plain $x^k$ instead of $(x-a)^k$, since $(x - 0) = x$.

How to build a Taylor polynomial:

1. Choose the center point $a$ and the degree $n$ of the polynomial.
2. Compute the derivatives $f(a), f'(a), f''(a), \ldots, f^{(n)}(a)$.
3. Plug each derivative into the formula, dividing by the corresponding factorial.

Quick example: Find the 3rd-degree Maclaurin polynomial for $e^x$.

Every derivative of $e^x$ is $e^x$, and $e^0 = 1$, so:

$P_3(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$

Plugging in $x = 0.5$: $P_3(0.5) = 1 + 0.5 + 0.125 + 0.0208 \approx 1.6458$, while the true value of $e^{0.5} \approx 1.6487$. That's pretty close with only four terms.

Interpretation of Taylor's Theorem

Taylor's theorem tells you exactly how the approximation error behaves. For a function $f(x)$ that is $(n+1)$ times differentiable on an interval containing $a$:

$f(x) = P_n(x) + R_n(x)$

where $R_n(x)$ is the remainder term (also called the Lagrange remainder):

$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$

Here $c$ is some value between $a$ and $x$. You typically don't know the exact value of $c$, but the theorem guarantees such a $c$ exists.

The remainder $R_n(x)$ represents the exact error between the true function value and the polynomial approximation. Two things shrink this error:

• Increasing $n$: More terms means a better fit, and the $(n+1)!$ in the denominator grows fast.
• Staying close to $a$: The factor $(x-a)^{n+1}$ gets smaller when $x$ is near the center point.

Taylor polynomials are especially valuable when:

• Direct evaluation of the function is difficult or impossible
• You know the derivatives at a point but don't have a closed-form expression
• You need a simpler expression for numerical computation (calculators use Taylor-type approximations internally for functions like $\sin(x)$)

Error Analysis in Taylor Series

Bounding the error is where Taylor's theorem becomes a practical tool, not just a theoretical one. The goal is to figure out how many terms you need to achieve a desired accuracy.

Steps to bound the error using the Lagrange error bound:

1. Identify the degree $n$ of your Taylor polynomial and the point $x$ where you're approximating.
2. Find the $(n+1)$th derivative $f^{(n+1)}(x)$.
3. Determine $M$, the maximum value of $|f^{(n+1)}(t)|$ for all $t$ between $a$ and $x$.
4. Apply the bound:

$|R_n(x)| \leq \frac{M}{(n+1)!}|x-a|^{n+1}$

Finding $M$ is often the trickiest step. For $\sin(x)$ and $\cos(x)$, all derivatives are bounded by 1, so $M = 1$ always works. For $e^x$ on $[0, x]$, the maximum of $e^t$ occurs at the right endpoint, so $M = e^x$.

Three factors that control the error size:

• Degree $n$: Higher degree means the $(n+1)!$ denominator grows rapidly, shrinking the bound.
• Distance $|x - a|$: The closer $x$ is to the center, the smaller the error. Approximating $e^{0.1}$ is far more accurate than $e^{2}$ with the same number of terms.
• Behavior of higher derivatives: If $|f^{(n+1)}|$ is large, the error can be large too.

Alternating series error bound: When your Taylor series happens to be alternating (like those for $\sin(x)$, $\cos(x)$, or $\ln(1+x)$), you can use a simpler bound:

$|R_n(x)| \leq |a_{n+1}|$

where $a_{n+1}$ is the first omitted term. This bound is often tighter than the Lagrange bound and easier to compute.

Convergence and Applications of Taylor Series

When you let $n \to \infty$, the Taylor polynomial becomes a Taylor series, an infinite power series:

$f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k$

This series doesn't necessarily converge for all $x$. You need to determine its radius of convergence $R$, typically using the Ratio Test or Root Test. The series converges for $|x - a| < R$ and diverges for $|x - a| > R$. At the endpoints $x = a \pm R$, you have to check convergence separately to find the full interval of convergence.

Within the interval of convergence, the Taylor series equals the original function (for the standard functions you'll encounter in this course). Some key Maclaurin series to know:

• $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$, converges for all $x$ ($R = \infty$)
• $\sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!}$, converges for all $x$ ($R = \infty$)
• $\cos(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!}$, converges for all $x$ ($R = \infty$)
• $\frac{1}{1-x} = \sum_{k=0}^{\infty} x^k$, converges for $|x| < 1$
• $\ln(1+x) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1} x^k}{k}$, converges for $-1 < x \leq 1$

Common applications of Taylor series include:

• Approximating function values when direct computation is impractical
• Evaluating difficult integrals by integrating the series term-by-term (e.g., $\int e^{-x^2} dx$ has no elementary antiderivative, but you can integrate its Taylor series)
• Solving differential equations by assuming a power series solution
• Analyzing local behavior of functions near a point (for instance, determining limits of indeterminate forms)