Infinite Series — AP Calculus BC
1. What is an Infinite Series? ★★☆☆☆ ⏱ 2 min
An infinite series is the sum of infinitely many terms of an infinite sequence, written formally as:
\sum_{n=1}^{\infty} a_n = a_1 + a_2 + a_3 + ...
The core question for any infinite series is whether it *converges*: if the partial sum $S_N = \sum_{n=1}^N a_n$ approaches a finite limit as $N \to \infty$, the series converges to that limit; otherwise, it diverges. Infinite series make up 10-15% of the AP Calculus BC exam, and are used for function approximation, numerical integration, and solving non-elementary differential equations.
2. Basic Convergence Tests ★★★☆☆ ⏱ 4 min
All tests below apply to series with non-negative terms, unless stated otherwise. AP examiners almost always require you to name the test you use and verify its conditions to earn full marks.
For series that resemble a known geometric or p-series, use one of the comparison tests:
- **Direct Comparison Test**: If $0 \leq a_n \leq b_n$ for all $n \geq N$ (for some fixed $N$): (1) if $\sum b_n$ converges, $\sum a_n$ converges; (2) if $\sum a_n$ diverges, $\sum b_n$ diverges.
- **Limit Comparison Test**: If $\lim_{n \to \infty} \frac{a_n}{b_n} = L$, where $0 < L < \infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.
3. Power Series & Intervals of Convergence ★★★☆☆ ⏱ 3 min
A power series is an infinite series of the form $\sum_{n=0}^{\infty} c_n (x - a)^n$, where $a$ is the center of the series and $c_n$ are constant coefficients. Every power series has a radius of convergence $R$: it converges for all $x$ with $|x-a| < R$, diverges for all $x$ with $|x-a|>R$, and convergence at endpoints $x=a \pm R$ must be tested separately.
To find the radius of convergence, use the Ratio Test:
\lim_{n \to \infty} \left| \frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n} \right| = |x - a| \cdot \lim_{n \to \infty} \left| \frac{c_{n+1}}{c_n} \right| = L
Set $L < 1$ and solve for $|x-a|$ to get $R$. The Ratio Test is always inconclusive at endpoints, so you must test these with other convergence tests.
4. Taylor & Maclaurin Series ★★★☆☆ ⏱ 3 min
A Taylor series is a power series representation of a function $f(x)$ centered at a point $a$, constructed using the derivatives of $f$ at $a$. The general form is:
\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n
Where $f^{(n)}(a)$ is the $n$th derivative of $f$ evaluated at $x=a$, and $f^{(0)}(a) = f(a)$. A **Maclaurin series** is a special case of a Taylor series centered at $a=0$, so its form simplifies to:
\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n
The $n$th-degree Taylor polynomial $P_n(x)$ is the partial sum of the first $n+1$ terms of the Taylor series, used to approximate $f(x)$ near $x=a$.
Memorize these four standard Maclaurin series to save time on the exam: you can modify them via substitution, differentiation, or integration to get series for other functions without computing derivatives from scratch:
- Geometric series: $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$, valid for $|x| < 1$
- Exponential function: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$, valid for all real $x$
- Sine function: $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, valid for all real $x$
- Cosine function: $\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}$, valid for all real $x$
5. Lagrange Error Bound ★★★★☆ ⏱ 3 min
The Lagrange error bound gives a guaranteed upper limit on the error of a Taylor polynomial approximation. If you approximate $f(x)$ with its $n$th-degree Taylor polynomial $P_n(x)$ centered at $a$, the remainder (error) $R_n(x) = f(x) - P_n(x)$ satisfies:
|R_n(x)| \leq \frac{M}{(n+1)!} |x - a|^{n+1}
Where $M$ is the maximum absolute value of the $(n+1)$th derivative of $f(t)$ for all $t$ between $a$ and $x$. AP examiners accept any valid upper bound for $M$, even if it is a conservative overestimate.
Common Pitfalls
Why: Students stop after calculating the radius of convergence, assuming endpoints are both convergent or divergent
Why: Students mix up test conditions and use direct/limit comparison for alternating series
Why: Students confuse the degree of the Taylor polynomial with the order of the derivative for the error term
Why: Students mix up the index of the alternating $(-1)^n$ term
Why: Students forget substitution changes the domain of convergence for finite-radius series