Topic 10.14 Notes – Finding Taylor or Maclaurin Series for a Function

What a Taylor and Maclaurin Series Are

A Taylor series expresses a function $f(x)$ as an infinite sum of powers of $(x-a)$, where $a$ is the center.

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

• $f^{(n)}(a)$ means the $n$th derivative evaluated at $x=a$.
• The powers are $(x-a)^n$, not just $x^n$ unless $a=0$.

A Maclaurin series is just a Taylor series centered at 0:

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

A Taylor polynomial of degree $n$ is the partial sum up to $n$. It’s a finite approximation; the full series is infinite.

The big picture: if you know all the derivatives at one point, you can rebuild the function (on its interval of convergence).

How to Construct a Taylor Series from the Definition

When you’re not given a known series, you go straight to the formula.

The Process

1. Compute derivatives: $f(a), f'(a), f''(a), \dots$
2. Plug into
   $$\frac{f^{(n)}(a)}{n!}(x-a)^n$$
3. Look for a pattern in $f^{(n)}(a)$.
4. Write the general term with sigma notation.
5. If they ask for a polynomial of degree $k$, stop at $n=k$.

Quick Example

Suppose $f(x)=e^{3x}$, centered at 0.

Derivatives:

• $f(x)=e^{3x}$
• $f'(x)=3e^{3x}$
• $f''(x)=9e^{3x}$

Pattern: $f^{(n)}(x)=3^n e^{3x}$

At 0:

$$f^{(n)}(0)=3^n$$

So the Maclaurin series is

$$\sum_{n=0}^{\infty}\frac{3^n x^n}{n!}$$

That pattern recognition step is what graders look for on FRQs.

Common mistakes:

• Forgetting the $n!$ in the denominator
• Plugging in $x$ instead of the center when evaluating derivatives
• Writing $x^n$ instead of $(x-a)^n$ when $a \neq 0$

Essential Maclaurin Series to Memorize

These are non-negotiable for BC.

Geometric Series

$$\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n \quad (|x|<1)$$

This is the foundation. Variations come from substitution:

• $\frac{1}{1+2x} = \sum (-2x)^n$
• Replace $x$ with an expression, then simplify carefully.

Exponential

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

Even better to remember:

$$e^{kx}=\sum_{n=0}^{\infty}\frac{k^n x^n}{n!}$$

All derivatives cycle back to itself. That’s why the pattern is clean.

Sine and Cosine

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

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

If you look at the first few Maclaurin polynomials for each function, the structure becomes obvious.

Sine and cosine with first few Maclaurin polynomials

Near $x=0$, the linear term $x$ closely matches $\sin x$, and adding higher odd powers improves the fit. For cosine, the constant term $1$ starts the approximation, and only even powers appear.

Patterns to lock in:

• Sine → odd powers
• Cosine → even powers
• Both alternate signs.

Logarithmic and Binomial

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

$$(1+x)^a=\sum_{n=0}^{\infty} \binom{a}{n} x^n$$

The binomial series works for non-integer $a$, which surprises people the first time they see it.

How to Build New Series from Known Ones

This is where most AP questions live.

Substitution

If you know the Maclaurin series for $\cos x$, then

$$\cos(4x)$$

means replace every $x$ with $4x$:

$$\sum (-1)^n \frac{(4x)^{2n}}{(2n)!}$$

Simplify powers completely. The AP loves asking for the coefficient of a specific power.

Algebraic Manipulation

You can:

• Multiply by constants
• Add or subtract known series
• Factor to match a known form

If you see something like
$$\frac{x}{1-x^2}$$ rewrite it first as
$$x \cdot \frac{1}{1-x^2}$$ then use the geometric idea.

Re-centering

If centered at $a$, every term becomes $(x-a)^n$, and derivatives are evaluated at $a$. Don’t just swap in $(x-a)$ at the end. The derivative values change too.

How the Series Represents the Function

A Taylor or Maclaurin series represents the function on its interval of convergence.

For geometric:

$$|x|<1$$

Substitutions change that inequality. For example, replacing $x$ with $3x$ gives $|3x|<1$, so $|x|<\frac{1}{3}$.

On tests, you’ll be asked to:

• Write first few terms
• Find a specific coefficient
• Recognize a function from its series
• State the interval of convergence

Students often forget that the interval matters just as much as the formula.