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Taylor & Maclaurin
Series

Learn to represent any smooth function as an infinite polynomial centered at a point: the key to approximation, computation, and deeper calculus.

AP Calculus BC

What Is a Power Series?

A power series is an infinite sum of terms involving increasing powers of \( (x - a) \):

The big idea: if we choose the coefficients correctly, this polynomial can perfectly match a function \( f(x) \) near \( x = a \).

Why Approximate with Polynomials?

Easy to Compute
Polynomials require only addition and multiplication: no transcendental operations needed.
Easy to Differentiate
Derivatives and integrals of polynomials are straightforward, term by term.
Arbitrarily Accurate
Adding more terms increases precision: 1 term gives a line, 3 terms a curve, and so on.

Visualizing Successive Approximations

The Taylor Series Formula

1
\( f^{(n)}(a) \): the \( n \)-th derivative of \( f \) evaluated at \( a \).
2
\( n! \): the factorial in the denominator controls the growth of each term.
3
\( (x - a)^n \): this factor ensures the polynomial is centered at \( x = a \).

Maclaurin Series: \( a = 0 \)

A Maclaurin series is simply a Taylor series centered at \( a = 0 \):

Every Maclaurin series is a Taylor series, but not every Taylor series is Maclaurin. If you are told "find the Taylor series at \( x = 0 \)," that is a Maclaurin series.

Taylor vs Maclaurin

Taylor Series Maclaurin Series
Center Any point \( a \) \( a = 0 \)
General Term \( \frac{f^{(n)}(a)}{n!}(x-a)^n \) \( \frac{f^{(n)}(0)}{n!}x^n \)
When Used Approximation near any \( a \) Approximation near \( 0 \)
Example \( \ln x \) at \( a = 1 \) \( e^x \) at \( a = 0 \)

How to Find a Taylor Series

1
Identify the center \( a \). If \( a = 0 \), it is a Maclaurin series.
2
Compute successive derivatives \( f(a),\; f'(a),\; f''(a),\; f'''(a), \ldots \)
3
Spot the pattern in \( f^{(n)}(a) \) and write the general term \( \frac{f^{(n)}(a)}{n!}(x-a)^n \).
4
Write in sigma notation: \( \displaystyle\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n \).
5
Determine the interval of convergence using the Ratio Test or known results.

Example 1: Maclaurin Series for \( e^x \)

Goal: Find the Maclaurin series for \( f(x) = e^x \).

1
Center: \( a = 0 \).
2
Derivatives: \( f^{(n)}(x) = e^x \) for all \( n \), so \( f^{(n)}(0) = 1 \).
3
General term: \( \dfrac{1}{n!}\,x^n \).

Example 2: Maclaurin Series for \( \sin x \)

Derivatives at 0 cycle: \( \sin 0 = 0,\; \cos 0 = 1,\; -\!\sin 0 = 0,\; -\!\cos 0 = -1,\;\ldots \) Only odd powers survive.

Example 3: Taylor Series for \( \ln x \) at \( a = 1 \)

We need derivatives of \( \ln x \) at \( x = 1 \):

0
\( f(1) = \ln 1 = 0 \)
1
\( f'(x) = \frac{1}{x} \implies f'(1) = 1 \)
2
\( f''(x) = -\frac{1}{x^2} \implies f''(1) = -1 \)
n
\( f^{(n)}(1) = (-1)^{n-1}(n-1)! \) for \( n \geq 1 \)

Must-Know Maclaurin Series

Function Series Convergence
\( e^x \) \( \displaystyle\sum \frac{x^n}{n!} \) All \( x \)
\( \sin x \) \( \displaystyle\sum \frac{(-1)^n x^{2n+1}}{(2n+1)!} \) All \( x \)
\( \cos x \) \( \displaystyle\sum \frac{(-1)^n x^{2n}}{(2n)!} \) All \( x \)
\( \frac{1}{1-x} \) \( \displaystyle\sum x^n \) \( |x| < 1 \)

Manipulating Known Series

Substitution
Replace \( x \) with an expression: e.g., \( e^{-x^2} \) uses the \( e^x \) series with \( x \to -x^2 \).
Differentiation
Differentiate term by term: \( \frac{d}{dx}\sin x = \cos x \) matches the \( \cos x \) series.
Integration
Integrate term by term: \( \int \frac{1}{1+x^2}\,dx = \arctan x \) gives its series.

Interval of Convergence

Every Taylor series converges in some interval \( (a - R,\; a + R) \) where \( R \) is the radius of convergence.

1
Use the Ratio Test: For a series \( \sum c_n(x-a)^n \), the radius is \( R = \displaystyle\lim_{n\to\infty}\left|\frac{c_n}{c_{n+1}}\right| \).
2
Check endpoints separately. The series may converge, diverge, or conditionally converge at \( x = a \pm R \).
3
Write the interval using parentheses or brackets depending on endpoint behavior.

Common Mistakes

✕ Forgetting \( n! \)
Writing \( f^{(n)}(a)(x-a)^n \) without dividing by \( n! \) gives wildly wrong coefficients.
✕ Wrong Center
Evaluating derivatives at \( 0 \) when the series is centered at \( a \neq 0 \), or using \( x^n \) instead of \( (x-a)^n \).
✕ Ignoring Endpoints
Finding \( R \) but forgetting to test \( x = a - R \) and \( x = a + R \) individually. Endpoints require their own convergence tests.

Question 1 of 5

What is the third-degree Maclaurin polynomial for \( f(x) = e^x \)?

Question 2 of 5

Which series represents the Maclaurin series for \( \cos x \)?

Question 3 of 5

What is the coefficient of \( (x-1)^3 \) in the Taylor series for \( \ln x \) centered at \( a = 1 \)?

Question 4 of 5

Find the Maclaurin series for \( e^{-x^2} \) through the \( x^4 \) term.

Question 5 of 5

What is the radius of convergence of the Maclaurin series for \( \dfrac{1}{1-x} \)?