Topic 10.15 Notes – Representing Functions as Power Series

What a Power Series Is

A power series centered at $r$ looks like

$$\sum_{n=0}^{\infty} a_n (x - r)^n$$

• $a_n$ are coefficients
• $r$ is the center
• It converges on an interval of convergence (sometimes including endpoints)

On BC, most problems are centered at 0, which makes them Maclaurin series:

$$\sum_{n=0}^{\infty} a_n x^n$$

Think of it as an infinite polynomial that behaves exactly like the function inside its interval of convergence.

The Series You Must Know Cold

These are your building blocks. If you recognize one of these hiding inside a problem, you’re in great shape.

1. Geometric Series

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

This is the most flexible one. Many rational functions get rewritten into this form.

2. Exponential

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

Converges for all real $x$.

3. Sine

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

4. Cosine

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

These also converge for all real $x$.

If you can write these from memory, you can derive almost everything in this topic.

The Four Ways to Build New Power Series

Every problem in this section is one of these moves.

1. Algebraic Manipulation

You can multiply, divide by constants, or add/subtract series.

If
$$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$$

Then
$$x^3 e^x = x^3 + x^4 + \frac{x^5}{2!} + \frac{x^6}{3!} + \dots$$

Multiplying by $x^k$ shifts all powers up by $k$.
Multiplying by a constant multiplies all coefficients.

Be careful rewriting the general term. If the original is $\frac{x^n}{n!}$, multiplying by $x^3$ gives $\frac{x^{n+3}}{n!}$.

2. Substitution

Replace $x$ with another expression everywhere.

From the geometric series:

$$\frac{1}{1-x} = \sum x^n$$

Replace $x$ with $4x$:

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

Now the interval changes:
$$|4x| < 1 \Rightarrow |x| < \frac{1}{4}$$

Always adjust the interval after substitution. That’s an easy place to lose a point.

3. Term-by-Term Differentiation

If
$$f(x) = \sum a_n x^n$$

then

$$f'(x) = \sum n a_n x^{n-1}$$

You differentiate exactly like a polynomial.

Example idea: differentiate the cosine series:

$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$$

Derivative:

$$-\sin x$$

You can literally see it appear term by term.

Important:

• The interval of convergence stays the same (check endpoints separately).
• This move is common on FRQs where they give you a series and ask for $f'(x)$ or $f''(x)$.

4. Term-by-Term Integration

If

$$f(x) = \sum a_n x^n$$

then

$$\int f(x)\,dx = \sum \frac{a_n}{n+1} x^{n+1} + C$$

This is how you get things like $\ln(1+x)$.

Start from geometric:

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

Integrate both sides and you get a series for $\ln(1+x)$.

This is a favorite AP move.

How to Handle “Find the First 4 Nonzero Terms and the General Term”

When that wording appears, do this mentally:

1. Identify the base series.
2. Rewrite the function to match it.
3. Apply one of the four moves.
4. Expand enough terms to clearly show the pattern.
5. Then write sigma notation.

Always expand first. Students lose points by jumping straight to sigma and messing up the pattern.

Common Traps

• Forgetting to substitute into the entire expression
• Messing up factorial shifts after differentiation
• Not adjusting the interval after substitution
• Losing the alternating $(-1)^n$
• Writing incorrect exponent patterns like $2n$ vs. $2n+1$

Remember what you’re doing here. You’re not inventing a brand-new series. You’re transforming a known one.