Unit 10.14 – Finding Taylor or Maclaurin Series for a Function BC ONLY

AP® Calculus BC | Building Infinite Polynomial Approximations

Why This Matters: **Taylor and Maclaurin series** let us turn complicated functions into infinite polynomials we can differentiate, integrate, and estimate easily. This is the key technique for finding series expansions and for tackling advanced calculus problems both on the AP® exam and in all of science and engineering.

🎯 Taylor & Maclaurin Series Formulas

General Taylor Series:

\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]

Where \(a\) is the center.

Maclaurin Series (centered at \(a = 0\)):

\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]

📝 Key Point: Taylor & Maclaurin are just the same process, Maclaurin is centered at zero.

📚 Three Main Methods to Find a Series

How to Find a Taylor/Maclaurin Series
Approach	Description / When Used	Typical Example
Method 1: Use Definition	Write derivatives, evaluate at center, build each term with formula	For any function if stuck!
Method 2: Use Known Series	Take a memorized series, substitute or manipulate (e.g., substitute, multiply, differentiate, integrate, shift center)	e.g. \(e^x, \sin x, \cos x, \frac{1}{1-x}\)
Method 3: Algebraic Manipulation	Rewrite function as a sum, product, derivative, or integral involving known series	\(e^{2x}\), \(\sin(3x^2)\), \(x^2\sin(x)\), differentiating or integrating term by term

⭐ Must-Know Maclaurin Series

Exponential:

\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \]

Sine:

\[ \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \]

Cosine:

\[ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \]

Geometric Series Function:

\[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \qquad (|x| < 1) \]

Natural Logarithm:

\[ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} \qquad (|x| < 1) \]

Binomial Series (general exponent):

\[ (1+x)^r = 1 + rx + \frac{r(r-1)}{2!}x^2 + \frac{r(r-1)(r-2)}{3!}x^3 + \cdots \]

📖 Worked Examples

Example 1: Finding Maclaurin Series for \(\sin(3x^2)\)

Start with: \(\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}\)

Substitute \(x \rightarrow 3x^2\):

\[ \sin(3x^2) = \sum_{n=0}^{\infty} \frac{(-1)^n (3x^2)^{2n+1}}{(2n+1)!} \]

\[ = \sum_{n=0}^{\infty} \frac{(-1)^n 3^{2n+1} x^{4n+2}}{(2n+1)!} \]

Example 2: Series for \(x^2 e^{x^3}\)

Start with: \(e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}\)

So \(e^{x^3} = \sum_{n=0}^{\infty} \frac{(x^3)^n}{n!} = \sum_{n=0}^{\infty} \frac{x^{3n}}{n!}\)

Multiply by \(x^2\):

\[ x^2 e^{x^3} = \sum_{n=0}^{\infty} \frac{x^{3n+2}}{n!} \]

Example 3: Taylor Series for \(f(x) = \frac{1}{x}\) at \(a = 1\)

Rewrite as geometric: \(\frac{1}{x} = \frac{1}{1 - (1-x)} = \sum_{n=0}^{\infty} (1-x)^n\), valid for |1-x| < 1

\[ \frac{1}{x} = \sum_{n=0}^{\infty} (1-x)^n \qquad (0 < x < 2) \]

Example 4: Integrate Known Series

Integrate the geometric series term-by-term to get Maclaurin for \(\ln(1+x)\):

\[ \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n \]

Integrate both sides:

\[ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} \]

Example 5: Taylor for \(f(x) = \arctan(x)\)

Derivative is \(f'(x)=\frac{1}{1+x^2}\), which has the Maclaurin series \(\sum_{n=0}^{\infty} (-1)^n x^{2n}\).

Integrate term-by-term to get:

\[ \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \]

💡 Tips & Tricks

Know the core series by heart! e.g., those in the box above.
Substitute, multiply, differentiate, or integrate: These let you use the core series for new functions easily.
Shifting center: For Taylor at \(a \neq 0\), use \((x-a)\) in place of \(x\).
Maclaurin = Taylor at 0: No shift needed, pure powers of \(x\).
If all else fails: Use the definition, compute first few terms by hand.

Memorize the core 5 Maclaurin series.
Check if your function is a multiple, derivative, integral, or composition involving these.
Substitute directly, expand and collect terms as needed.
For rational functions: try geometric series first.
For trigonometric/exponential: substitute into memorized sine/cos/e^x.
Show pattern clearly for non-integer coefficients.

❌ Common Mistakes to Avoid

Forgetting to substitute for entire powers (e.g., \(x^2\) vs \(x\)).
Not checking the interval of convergence (pick up quick points!).
Misapplying geometric series—use only when \(|r|<1\).
Leaving series in unsimplified form instead of writing an explicit pattern.
Saying Maclaurin when you mean Taylor (wrong center).
Forgetting factorials in denominators.

📝 Practice Problems

Write the Taylor or Maclaurin series (first 4 nonzero terms):

\(\sin(2x)\)
\(x^2 e^{x^3}\)
\(\cos(x^2)\)
\(\frac{1}{1-2x}\)
\(\arctan(x^2)\)

Answers:

\(2x - \frac{(2x)^3}{6} + \frac{(2x)^5}{120} - \frac{(2x)^7}{5040}\)
\(x^2 + x^5 + \frac{x^8}{2} + \frac{x^{11}}{6}\)
\(1 - x^4/2 + x^8/24 - x^{12}/720\)
\(1 + 2x + 4x^2 + 8x^3\)
\(x^2 - x^6/3 + x^{10}/5 - x^{14}/7\)

⚡ Quick Reference

Taylor: \(\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\)
Maclaurin: \(\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n\)
Core series:
- \(e^x, \sin x, \cos x, \frac{1}{1-x}, \ln(1+x)\)
Most shortcuts: Substitute, differentiate, integrate!

Master Taylor & Maclaurin Series! You can build the series for almost any differentiable function using the right combination of: (1) writing the definition and calculating derivatives, (2) substituting and manipulating the core memorized series, and (3) using algebraic tricks (differentiation/integration, multiplication, composition, shifting center). Start with known patterns, then manipulate—the fastest way to find any series expansion on the AP® exam!