Introduction to Maclaurin Series
Welcome to one of the most powerful tools in higher-level calculus! Have you ever wondered how your calculator actually "knows" the value of \( \sin(0.5) \) or \( e^{0.2} \)? It doesn't have an infinite list of numbers stored in its memory. Instead, it uses polynomials to approximate these complex functions. In this chapter, we will learn how to turn complicated functions like trigonometry and logarithms into simple, infinite sums of \( x \), \( x^2 \), \( x^3 \), and so on. This is what we call a Maclaurin series.
The Maclaurin series is specifically an expansion of a function centered at \( x = 0 \). It allows us to treat curvy, complex functions as if they were just long polynomials, which makes them much easier to differentiate, integrate, and solve.
The General Maclaurin Formula
To create a Maclaurin series for a function \( f(x) \), we need the function to be "well-behaved" (differentiable) at \( x = 0 \). The general formula found in your formula booklet is:
\( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots + \frac{f^{(n)}(0)}{n!}x^n + \dots \)
Breaking down the notation:
1. \( f(0) \): The value of the function when \( x = 0 \).
2. \( f'(0) \), \( f''(0) \): The first and second derivatives evaluated at \( x = 0 \).
3. \( n! \): This is "n factorial" (e.g., \( 3! = 3 \cdot 2 \cdot 1 = 6 \)).
4. \( f^{(n)}(0) \): The \( n^{th} \) derivative evaluated at \( x = 0 \).
Key Takeaway: The more terms we include in our series, the more accurate our "polynomial approximation" becomes compared to the original function.
The "Standard" Maclaurin Series
In the IB HL curriculum, you are expected to know and use several specific series. While these are in your formula booklet, being familiar with them will save you a lot of time!
1. Exponential Function:
\( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \)
Memory Aid: This is the easiest one—it includes every power of \( x \) and every factorial!
2. Sine and Cosine Functions:
\( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \) (Only odd powers; it is an odd function)
\( \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \) (Only even powers; it is an even function)
Tip: Notice the signs alternate between \( + \) and \( - \).
3. Logarithm and Arctan:
\( \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots \)
\( \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \dots \)
Common Mistake Alert: Students often forget that these two do not use factorials in the denominators! They just use the regular integers \( 2, 3, 4 \dots \) or \( 3, 5, 7 \dots \)
4. The Binomial Series:
For rational \( p \):
\( (1+x)^p = 1 + px + \frac{p(p-1)}{2!}x^2 + \frac{p(p-1)(p-2)}{3!}x^3 + \dots \)
Building New Series from Old Ones
You don't always have to start from the general formula. You can manipulate the standard series to find series for more complex functions.
1. Substitution:
If you know \( e^x \), you can find \( e^{x^2} \) by simply replacing every \( x \) in the series with \( x^2 \).
Example: \( e^{x^2} = 1 + (x^2) + \frac{(x^2)^2}{2!} + \dots = 1 + x^2 + \frac{x^4}{2} + \dots \)
2. Multiplication by a variable:
To find the series for \( x \sin x \), take the series for \( \sin x \) and multiply every term by \( x \).
Example: \( x(x - \frac{x^3}{3!} + \dots) = x^2 - \frac{x^4}{6} + \dots \)
3. Differentiation and Integration:
Since a Maclaurin series is just a polynomial, you can differentiate it term-by-term. For example, if you differentiate the series for \( \sin x \), you will actually get the series for \( \cos x \)!
Maclaurin Series from Differential Equations
Sometimes, the IB will give you a differential equation (e.g., \( \frac{dy}{dx} = x + y^2 \)) and an initial condition like \( y(0) = 1 \). They will then ask you to find the first few terms of the Maclaurin series for \( y \).
Step-by-Step Process:
1. Find \( f(0) \): Use the given initial condition (e.g., if \( y = 1 \) when \( x = 0 \), then \( f(0) = 1 \)).
2. Find \( f'(0) \): Plug \( x = 0 \) and your value for \( y(0) \) into the differential equation.
3. Find \( f''(0) \): Differentiate the original differential equation with respect to \( x \) (use implicit differentiation if needed!) and plug in your known values for \( x, y, \) and \( y' \).
4. Assemble: Plug these values into the general Maclaurin formula.
Don't worry if this seems tricky at first! The key is keeping your derivatives organized. If you need a refresher on implicit differentiation, check the "Implicit differentiation and related rates" chapter.
Using Maclaurin Series to Evaluate Limits
Maclaurin series are a fantastic alternative to l'Hopital's Rule when finding limits of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). (See the chapter on "Limits and l'Hopital's rule" for the alternative method).
How to do it: Replace the trigonometric or exponential parts of the fraction with their Maclaurin series. Simplify the fraction by canceling out terms. The limit as \( x \to 0 \) will usually become obvious once the "problematic" \( x \) terms are canceled.
Example: Find \( \lim_{x \to 0} \frac{\sin x - x}{x^3} \).
Substitute the series for \( \sin x \):
\( \lim_{x \to 0} \frac{(x - \frac{x^3}{6} + \dots) - x}{x^3} = \lim_{x \to 0} \frac{-\frac{x^3}{6}}{x^3} = -\frac{1}{6} \).
Summary of Key Points
1. Definition: Maclaurin series represent functions as infinite polynomials centered at \( x = 0 \).
2. Standard Series: Know the expansions for \( e^x, \sin x, \cos x, \ln(1+x), \arctan x, \) and \( (1+x)^p \).
3. Factorials: Remember factorials go with \( e, \sin, \) and \( \cos \), but not with \( \ln \) and \( \arctan \).
4. Manipulation: You can substitute, multiply, differentiate, and integrate series to create new ones.
5. Limits: Use series to simplify complex fractions when finding limits as \( x \) approaches \( 0 \).
Quick Review: If you are asked for the series of \( \cos(3x) \), start with the series for \( \cos x \) and replace every \( x \) with \( (3x) \). Don't forget to square or cube the 3 as well!