Taylor & Maclaurin Series

Taylor & Maclaurin Series — The General Formula

Part 1 of 7 — Taylor Polynomial Construction

Taylor Series Centered at $x = c$

$\boxed{f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x - c)^n}$

$= f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots$

Maclaurin Series (Special Case: $c = 0$)

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

Taylor Polynomials

The $n$th-degree Taylor polynomial is the partial sum:

$T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(c)}{k!}(x-c)^k$

AP Tip: "Write the $n$th-degree Taylor polynomial" means $T_n(x)$. "Write the first four nonzero terms of the Taylor series" may give a higher-degree polynomial.

Example: Taylor Series for $e^x$ at $c = 0$

$f(x) = e^x \implies f^{(n)}(x) = e^x \implies f^{(n)}(0) = 1$ for all $n$

$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots$

Example: Taylor Series for $\sin x$ at $c = 0$

Pattern repeats with period 4: $0, 1, 0, -1, 0, 1, 0, -1, \ldots$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!}$

Taylor Polynomial Basics

Building Taylor Polynomials

Derivative Extraction

Summary

• Taylor series: $\sum f^{(n)}(c)(x-c)^n/n!$
• Maclaurin series: Taylor at $c = 0$
• $T_n(x)$ = partial sum through degree $n$
• Know the difference between "degree $n$" and "first $k$ nonzero terms"

Next: Part 2 — Computing Taylor Series from Scratch.

Taylor & Maclaurin Series — Computing from Scratch

Part 2 of 7 — Derivative Tables & Non-Zero Centers

Method: Derivative Table

To build the Taylor series at $c$:

1. Compute $f(c), f'(c), f''(c), f'''(c), \ldots$
2. Form coefficients $a_n = f^{(n)}(c)/n!$
3. Write $\sum a_n(x-c)^n$

Example: $f(x) = \sqrt{x}$ at $c = 4$

$\sqrt{x} \approx 2 + \frac{1}{4}(x-4) - \frac{1}{64}(x-4)^2 + \frac{1}{512}(x-4)^3 - \cdots$

AP Tip: This is the method for functions NOT in the "big six" list. You compute derivatives until a pattern emerges or until you have enough terms.

Taylor Series at Non-Zero Centers

Example: $e^x$ centered at $c = 1$

$f^{(n)}(1) = e$ for all $n$.

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

Example: $\sin x$ centered at $c = \pi/2$

$\sin x = 1 - \frac{(x - \pi/2)^2}{2!} + \frac{(x-\pi/2)^4}{4!} - \cdots$

Notice this looks like $\cos(x - \pi/2)$, which makes sense since $\sin x = \cos(x - \pi/2)$!

Computing Taylor Series

Derivative Computation

Taylor at Non-Zero Center

Summary

• Build Taylor series via derivative tables
• Non-zero centers: evaluate at $c$, use $(x-c)^n$
• Look for patterns in derivatives (cyclic, factorial, powers)
• On the AP exam, you typically need 3-4 terms, not the general formula

Next: Part 3 — Known Series and Manipulation Techniques.

Taylor & Maclaurin — Known Series & Manipulation

Part 3 of 7 — Using Known Series to Build New Ones

The Big Six (Must Memorize)

Manipulation Toolkit

AP Tip: Building from known series is MUCH faster than computing derivatives from scratch. Always try this first.

Worked Examples

Example 1: $xe^{-x}$

$e^{-x} = \sum \frac{(-x)^n}{n!} = \sum \frac{(-1)^n x^n}{n!}$

$xe^{-x} = \sum_{n=0}^\infty \frac{(-1)^n x^{n+1}}{n!} = x - x^2 + \frac{x^3}{2} - \frac{x^4}{6} + \cdots$

Example 2: $\cos^2 x$ (using identity)

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

$= \frac{1}{2} + \frac{1}{2}\left[1 - 2x^2 + \frac{2x^4}{3} - \cdots\right]$

Wait: $\cos(2x) = 1 - (2x)^2/2! + (2x)^4/4! - \cdots = 1 - 2x^2 + 2x^4/3 - \cdots$

$\cos^2 x = 1 - x^2 + x^4/3 - \cdots$

Key Insight: Using trig identities + known series is often the fastest approach.

Manipulation Practice

Series Construction

Quick Coefficient

Summary

• Always try manipulating known series before computing derivatives
• Substitution, multiplication, and differentiation/integration are the main tools
• Trig identities can simplify series construction
• Only even/odd powers? Pay attention to symmetry

Next: Part 4 — Taylor's Theorem and the Remainder.

Taylor & Maclaurin — Taylor's Theorem & Remainder

Part 4 of 7 — The Lagrange Error Bound

Taylor's Theorem

If $f$ has $(n+1)$ continuous derivatives, then:

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

where $R_n(x) =$ the remainder (error of approximation).

The Lagrange Error Bound

$\boxed{|R_n(x)| \le \frac{M \cdot |x - c|^{n+1}}{(n+1)!}}$

where $M = \max_{t}|f^{(n+1)}(t)|$ on the interval between $c$ and $x$.

Comparison of Error Bounds

AP Tip: Use the AST error bound when the series alternates — it's simpler. Use Lagrange when it doesn't alternate or when specifically asked.

Example: Bound the Error of $e^x \approx T_3(x)$ at $x = 0.5$

$T_3(0.5) = 1 + 0.5 + 0.125 + 0.0208\overline{3} = 1.6458\overline{3}$

For the Lagrange bound: $f^{(4)}(x) = e^x$

$M = \max_{0 \le t \le 0.5} e^t = e^{0.5} \approx 1.649$

$|R_3(0.5)| \le \frac{1.649 \cdot (0.5)^4}{4!} = \frac{1.649 \cdot 0.0625}{24} \approx 0.00429$

Actual: $e^{0.5} \approx 1.6487$, error $\approx 0.0029$. The bound ($0.0043$) is correct and conservative.

Common $M$ Values

Lagrange Error Practice

Error Bound Decisions

Lagrange Computation

Summary

• Lagrange Error: $|R_n(x)| \le M|x-c|^{n+1}/(n+1)!$
• Find $M$ by bounding $|f^{(n+1)}|$ on the interval
• For $\sin/\cos$: $M = 1$ always
• For $e^x$: $M = e^{|x|}$ (or use crude bound like $3$)
• Use AST when series alternates (it's tighter and easier)

Next: Part 5 — AP FRQ Strategies for Taylor Series.

Taylor & Maclaurin — AP FRQ Strategies

Part 5 of 7 — Exam Techniques

The FRQ Taylor Series Question

This appears on virtually EVERY BC exam. The typical structure:

Part (a): Write the first 4 nonzero terms and the general term of the Taylor/Maclaurin series for $f$.

Part (b): Find the interval/radius of convergence.

Part (c): Use the series to approximate a value or integral.

Part (d): Show the approximation has error less than some bound.

Part (a) Strategy

AP Tip: "General term" means write $\sum$ notation with $n$. This is where students lose the most points — verify your general term by checking it produces the first few terms correctly.

Part (b): Interval of Convergence

Always use Ratio Test → test endpoints.

Write your answer as an interval with proper notation: $[-1,1)$, not "$-1$ to $1$."

Part (c): Approximation

Substitute the given value into your series: $\sin(0.5) \approx 0.5 - (0.5)^3/6 + (0.5)^5/120 = 0.5 - 0.02083 + 0.00026 = 0.47943$

Part (d): Error Bound

Choose between:

• AST Error Bound if the series alternates (simpler!)
• Lagrange Error Bound if not alternating or specifically asked

Template for AST: "Since the series is alternating with decreasing terms converging to $0$, the error is bounded by the first omitted term: $|R| \le |a_{N+1}| = \ldots < \epsilon$."

Template for Lagrange: "By Taylor's theorem, $|R_n(x)| \le M|x-c|^{n+1}/(n+1)!$ where $M = \max|f^{(n+1)}| = \ldots$"

Scoring Insight

Each part is typically worth 2-3 points. Justification language matters — use precise mathematical statements.

AP Question Types

FRQ Decisions

FRQ Practice

Summary

• The Taylor FRQ has a predictable structure: series → IOC → approx → error
• Use known series when possible; compute derivatives as last resort
• For error bounds: AST when alternating, Lagrange otherwise
• Verify your general term reproduces the terms you wrote
• Show ALL work — the AP graders need to see your reasoning

Next: Part 6 — Problem-Solving Workshop.

Taylor & Maclaurin — Problem-Solving Workshop

Part 6 of 7 — Mixed Practice

Workshop Focus

This workshop covers the full range of Taylor series tasks:

• Computing series from scratch
• Manipulating known series
• Finding intervals of convergence
• Error bound calculations
• Integrating/differentiating series

Workshop Problems

Series Identification

Derivative from Series

Workshop Takeaways

• Derivative table method for unfamiliar functions (like $\tan x$)
• Binomial series for $(1+x)^p$ when $p$ is not a positive integer
• Series evaluation by identifying the function
• Integration of series for functions with no elementary antiderivative

Next: Part 7 — Comprehensive Review.

Taylor & Maclaurin — Comprehensive Review

Part 7 of 7 — Complete Topic Review

Master Reference

The Six Essential Series

$\boxed{\frac{1}{1-x},\quad e^x,\quad \sin x,\quad \cos x,\quad \ln(1+x),\quad \arctan x}$

Key Fact: Taylor/Maclaurin series is the single most heavily tested BC topic. Expect 4+ MC questions and a full FRQ.

Comprehensive Review MC

More Review

Final Drill

Final Challenge

Taylor & Maclaurin — Complete Summary

You've mastered:

• Taylor formula — $\sum f^{(n)}(c)(x-c)^n/n!$
• Computing from scratch — derivative tables
• Known series manipulation — substitution, products, differentiation, integration
• Lagrange error bound — $M|x-c|^{n+1}/(n+1)!$
• AP FRQ strategies — the 4-part structure
• Coefficient-derivative connection — $f^{(n)}(c) = n! \cdot a_n$

Up Next: Lagrange Error Bound — deeper exploration of remainder estimation.