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Unit 10.15 – Representing Functions as Power Series BC ONLY

AP® Calculus BC | Turning Functions into Infinite Polynomials for Powerful Approximations

Why This Matters: **Representing functions as power series** is a cornerstone of BC Calculus and applied math. It lets us write any "nice" function as an infinite polynomial, allowing term-by-term integration, differentiation, and easy approximations in science and engineering.

🎯 The General Power Series

General Form:
\[ f(x) = \sum_{n=0}^{\infty} c_n(x-a)^n \]

where \(a\) is the center, \(c_n\) are coefficients

Maclaurin Series (centered at 0):
\[ f(x) = \sum_{n=0}^{\infty} c_n x^n \]
Taylor Series (centered at \(a\)):
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]
📝 Key Insight: Any function that is analytic can (often) be represented as a power series within its interval of convergence!

🔎 Three Main Methods for Representing Functions as Power Series

How to Convert a Function to a Power Series
Method How It Works & Tips Typical Example
1. By Definition Write all derivatives, evaluate at center, build terms using Taylor/Maclaurin formula f(x) = sin(x), ln(1+x)
2. Manipulate a Known Series Substitute, multiply, differentiate, integrate, or shift center. Fastest for most functions!
Tricks:
  • Substitute (e.g. x → 2x, x², -x, etc.)
  • Multiply series by powers or x
  • Differentiate/integrate series term by term
  • Shift center: x → (x–a)
 e.g., \( e^{3x^2} \), \( \sin(x^2) \), \( \arctan(2x) \)
3. Algebraic and Calculus Tricks Factor, partial fractions, composition, use geometric/known series
Integrate/differentiate if stuck! 		e.g., Integrate \( 1/(1+x^2) \) to get arctangent series

⭐ Core Maclaurin Series to Use

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/Basic Rational:
\[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n\qquad (|x|<1) \]
Natural Logarithm:
\[ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^n}{n}\qquad (|x|<1) \]
Arctangent:
\[ \arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \]

📖 Worked Examples

Example 1: Substitute and Shift

Represent \(f(x) = \frac{1}{1-3x}\) as a power series.

Start from core geometric: \( \frac{1}{1-x} = \sum x^n \).

Substitute \(x \rightarrow 3x\):

\[ \frac{1}{1-3x} = \sum_{n=0}^{\infty} (3x)^n = \sum_{n=0}^{\infty} 3^n x^n, \qquad (|3x|<1, |x|<1/3) \]

Example 2: Integrate a Known Series

Represent \(f(x) = \ln(1+x)\).

Start with \( \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^n x^n \).

Integrate term by term:

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

Example 3: Differentiate a Known Series

Represent \(f(x) = \frac{1}{(1-x)^2}\) as a power series.

Differentiate the geometric series:

\[ \frac{d}{dx}\left(\frac{1}{1-x}\right) = \sum_{n=1}^{\infty} n x^{n-1} \]
\[ \frac{1}{(1-x)^2} = \sum_{n=1}^{\infty} n x^{n-1} \]

Example 4: Compose: Cosine of \(3x^2\)

Represent \(f(x) = \cos(3x^2)\) as a Maclaurin series.

From core: \( \cos x = \sum (-1)^n x^{2n}/(2n)! \). Substitute \(x\rightarrow 3x^2\):

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

Example 5: Maclaurin for \(e^{-x^2}\)

Represent \(f(x) = e^{-x^2}\) as a power series.

From core: \( e^x = \sum x^n/n! \).

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

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

💡 Tips and Shortcuts

  • Start with a memorized series.
  • Substitute, multiply, differentiate, integrate, or compose as needed.
  • Circle/stick with the SAME center as in the original series unless shifting is required.
  • Always state the interval of convergence if possible.
  • For rational functions, start with geometric; expressions like \(\sin(ax)\) or \(e^{bx^2}\) use direct substitution.
  • Check if coefficients need simplifying or if you need to re-index.
  1. Identify a matching memorized series.
  2. Write out the first several terms to spot the pattern.
  3. Rewrite the target function (substitute, shift variable, etc.).
  4. Adjust all exponents and coefficients to match new function.
  5. STATE THE INTERVAL OF CONVERGENCE based on transformation.
  6. Simplify to neat, bold, clear answer.

❌ Common Mistakes

  • Forgetting to check and convert the interval of convergence after substitution.
  • Not multiplying out the powers properly when substituting.
  • Combining the wrong exponents or coefficients in your answer.
  • Forgetting to include the center (e.g., using Maclaurin when Taylor is required).
  • Implicitly assuming the interval always remains |x| < 1.
  • Leaving powers in unsimplified form (e.g., \((2x)^n\) not expanded).

📝 Practice Problems

Write a power series for each function (first 3 nonzero terms):

  1. \(\frac{1}{1+2x}\)
  2. \(e^{3x}\)
  3. \(\sin(2x^3)\)
  4. \(\arctan(4x)\)
  5. \(\ln(1-x^2)\)

Answers:

  1. \(1 - 2x + 4x^2\)
  2. \(1 + 3x + \frac{9x^2}{2}\)
  3. \(2x^3 - \frac{8x^9}{3} + \frac{32x^{15}}{15}\)
  4. \(4x - \frac{64x^3}{3} + \frac{1024x^5}{5}\)
  5. \(-x^2 - \frac{x^4}{2} - \frac{x^6}{3}\)

⚡ Quick Reference

  • General Power Series: \(\sum_{n=0}^\infty c_n (x-a)^n\)
  • Maclaurin: \(\sum_{n=0}^\infty c_n x^n\)
  • Taylor: \(\sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n\)
  • Powers/compositions/substitutions: Plug into memorized series, adjust exponents and intervals.
  • Interval changes: Make sure you update for substitutions!

Master Power Series Representation! The fastest approach on the AP® exam is to spot a memorized series and make appropriate substitutions (x → ax, x², etc.), then adjust your answer. For harder functions, integrate, differentiate, or combine known series. Always box your final answer and state the interval if required!