On the AP Calculus BC exam, you are expected to know the Maclaurin series for several common functions. Memorizing these series will save you time and help you recognize patterns when manipulating series.
🎯 In this section you will learn
- The Maclaurin series for the six most important functions
- Their intervals of convergence
- How to use these known series to find series for related functions
📌 The Essential Maclaurin Series
| Function | Maclaurin Series | Interval of Convergence |
|---|
| $e^x$ | $\displaystyle \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$ | $(-\infty, \infty)$ |
| $\sin x$ | $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$ | $(-\infty, \infty)$ |
| $\cos x$ | $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$ | $(-\infty, \infty)$ |
| $\frac{1}{1-x}$ | $\displaystyle \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + \cdots$ | $(-1, 1)$ |
| $\ln(1+x)$ | $\displaystyle \sum_{n=1}^\infty \frac{(-1)^{n-1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ | $(-1, 1]$ |
| $\arctan x$ | $\displaystyle \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \cdots$ | $[-1, 1]$ |
💡 Memory Tricks
- $e^x$: "all derivatives are $e^x$, so all coefficients are $1/n!$"
- $\sin x$ and $\cos x$: "sine has only odd powers, cosine has only even powers, signs alternate"
- $\frac{1}{1-x}$: "geometric series with ratio $x$"
- $\ln(1+x)$: "integral of $\frac{1}{1+x}$"
- $\arctan x$: "integral of $\frac{1}{1+x^2}$"
Find the Maclaurin series for $f(x) = e^{2x}$.
①
Use known series for $e^x$
$e^u = \sum_{n=0}^\infty \frac{u^n}{n!}$
②
Substitute $u = 2x$
$e^{2x} = \sum_{n=0}^\infty \frac{(2x)^n}{n!} = \sum_{n=0}^\infty \frac{2^n x^n}{n!}$
Find the Maclaurin series for $f(x) = \sin(x^2)$.
- $\sin u = \sum_{n=0}^\infty \frac{(-1)^n u^{2n+1}}{(2n+1)!}$
- Let $u = x^2$: $\sin(x^2) = \sum_{n=0}^\infty \frac{(-1)^n (x^2)^{2n+1}}{(2n+1)!} = \sum_{n=0}^\infty \frac{(-1)^n x^{4n+2}}{(2n+1)!}$
Find the Maclaurin series for $f(x) = \frac{x}{1-x}$.
- $\frac{1}{1-x} = \sum_{n=0}^\infty x^n$
- Multiply by $x$: $\frac{x}{1-x} = \sum_{n=0}^\infty x^{n+1} = \sum_{n=1}^\infty x^n$
Find the Maclaurin series for $f(x) = \ln(1+3x)$.
- $\ln(1+u) = \sum_{n=1}^\infty \frac{(-1)^{n-1} u^n}{n}$
- Let $u = 3x$: $\ln(1+3x) = \sum_{n=1}^\infty \frac{(-1)^{n-1} (3x)^n}{n} = \sum_{n=1}^\infty \frac{(-1)^{n-1} 3^n x^n}{n}$
Find the Maclaurin series for $f(x) = \arctan(x^2)$.
- $\arctan u = \sum_{n=0}^\infty \frac{(-1)^n u^{2n+1}}{2n+1}$
- Let $u = x^2$: $\arctan(x^2) = \sum_{n=0}^\infty \frac{(-1)^n (x^2)^{2n+1}}{2n+1} = \sum_{n=0}^\infty \frac{(-1)^n x^{4n+2}}{2n+1}$
⚠️ Important Notes
- Memorize these six series — they appear frequently on the AP exam.
- Know the intervals of convergence, especially for $\ln(1+x)$ ($-1,1]$) and $\arctan x$ ($[-1,1]$).
- Be careful with substitution: When substituting $u = cx^k$, the interval of convergence changes accordingly (e.g., $|cx^k| < 1$).
🔍 Key Takeaways
- Memorize Maclaurin series for $e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$, $\ln(1+x)$, $\arctan x$.
- Use substitution to find series for related functions.
- Intervals of convergence are important — check endpoints for $\ln$ and $\arctan$.
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