Calculus BC · Infinite Sequences and Series (Unit 10) · 14 min read · Updated 2026-05-11
Representing functions as power series — AP Calculus BC
AP Calculus BC · Infinite Sequences and Series (Unit 10) · 14 min read
1. Geometric Power Series Representations★★☆☆☆⏱ 4 min
The foundation for all power series representations is the infinite geometric series formula you already know. For $|r| < 1$, the sum of the infinite series is $\sum_{n=0}^\infty r^n = \frac{1}{1-r}$. If we let $r = g(x)$, we can write any rational function rearranged to the form $\frac{1}{1-g(x)}$ directly as a power series, convergent when $|g(x)| < 1$.
Exam tip: Always start by rewriting the denominator to get 1 as the constant term, matching the $1-r$ form. If you factor incorrectly, you will get wrong coefficients for every term of the series.
2. Term-by-Term Differentiation of Power Series★★★☆☆⏱ 4 min
Within the interval of convergence (excluding possibly endpoints), power series can be differentiated term-by-term just like finite polynomials. This property lets us build new power series from known existing representations. If $f(x) = \sum_{n=0}^\infty c_n (x-a)^n$ has radius of convergence $R$, the derivative has the same radius of convergence, but you must recheck convergence at endpoints.
f'(x) = sum_{n=1}^infty n c_n (x-a)^{n-1}
Exam tip: When you differentiate a power series, the constant term disappears, so the starting index always shifts from $n=0$ to $n=1$. Forgetting to adjust the starting index is a common multiple-choice trap answer.
3. Term-by-Term Integration of Power Series★★★☆☆⏱ 4 min
Like differentiation, power series can be integrated term-by-term within their interval of convergence. This is an extremely useful technique that lets us find power series for transcendental functions that are integrals of rational functions, such as $\ln(1+x)$ and $\arctan x$.
int f(x) dx = C + sum_{n=0}^infty frac{c_n}{n+1} (x-a)^{n+1}
The radius of convergence stays the same as the original series, but you must recheck convergence at endpoints. The constant of integration $C$ is found by substituting $x=a$ (the center of the series) into the function.
Exam tip: Don't forget to solve for the constant of integration $C$ when integrating a power series. For Maclaurin series centered at 0, $C = f(0)$, which is almost always 0 for common functions like $\ln(1+x)$ or $\arctan x$.
4. AP-Style Concept Check★★★★☆⏱ 2 min
Common Pitfalls
Why: Students know the radius of convergence stays the same, so they incorrectly assume the entire interval stays the same, but convergence behavior can change at endpoints.
Why: Students misidentify $r$ in the geometric series formula, forgetting the denominator is $1-r$, so $+x$ means $r=-x$.
Why: Students rush the substitution when shifting the starting index of the series.
Why: Students forget that factoring a constant from the denominator moves its reciprocal to the outside of the fraction.
Why: Students confuse the variable of differentiation $x$ with the index $n$, so they unnecessarily modify the coefficients.