AP Calculus BC: Infinite Series

Infinite series sit at the heart of AP Calculus BC because they connect limits, sequences, integrals, and derivatives into one powerful idea: you can represent complicated functions using infinitely many simpler terms. They are also the most conceptually demanding unit for many students, not because any single test is hard in isolation, but because success depends on choosing the right tool quickly and justifying it cleanly.

This article lays out a systematic way to analyze convergence, then builds into power series and Taylor and Maclaurin series, which are essential both for the exam and for genuine mathematical understanding.

What an Infinite Series Is (and What It Is Not)

An infinite series is a sum of infinitely many terms: $$\sum _{n=1}^{\infty }{a}_n.$$

A series is related to, but different from, the sequence of its terms $\{a_n\}$ and the sequence of its partial sums: $$S_N=\sum _{n=1}^N{a}_n.$$

The series converges if the partial sums $S_N$ approach a finite limit as $N\to \infty$. It diverges otherwise.

A necessary condition: the nth-term test

If $\sum a_n$ converges, then $a_n\to 0$. This gives the fastest divergence check:

• If ${\lim }_{n\to \infty }{a}_n\ne 0$ or the limit does not exist, then $\sum a_n$ diverges.

Be careful: $a_n\to 0$ does not guarantee convergence. It is necessary, not sufficient.

A Systematic Approach to Convergence Tests

AP Calculus BC expects you to recognize patterns and apply the most efficient convergence test with clear reasoning. A practical decision path looks like this:

Step 1: Identify “special” series immediately

Some series are so standard that you should recognize them without testing.

Geometric series

$$\sum _{n=0}^{\infty }a{r}^n$$ converges to $\frac{a}{1-r}$ if $|r|<1$, and diverges if $|r|\ge 1$.

Many problems hide the ratio in a form like ${\left(\frac{3}{2}\right)}^n$, ${\left(-\frac{1}{5}\right)}^n$, or $2^n/3^n={\left(\frac{2}{3}\right)}^n$.

p-series

$$\sum _{n=1}^{\infty }\frac{1}{n^p}$$ converges if $p>1$ and diverges if $p\le 1$.

If the series is a constant multiple of a p-series, it behaves the same way.

Step 2: Look for comparison opportunities

If the series resembles a p-series or a geometric series but is not exactly one, comparison tests are often best.

Direct Comparison Test

If $0\le a_n\le b_n$ eventually:

• If $\sum b_n$ converges, then $\sum a_n$ converges.
• If $\sum a_n$ diverges, then $\sum b_n$ diverges.

This works well when inequalities are obvious, such as comparing $\frac{1}{n^2+1}$ to $\frac{1}{n^2}$.

Limit Comparison Test

If $a_n>0$, $b_n>0$, and $$\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=c$$ with $0<c<\infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge.

This is especially useful for rational expressions in $n$ like $\frac{n^2+5}{n^3-1}$, where leading terms determine behavior.

Step 3: Check for alternating structure

If terms alternate signs, you may be able to use the Alternating Series Test (Leibniz).

A series of the form $$\sum (-1{)}^n{b}_n\text{or}\sum (-1{)}^{n+1}{b}_n$$ converges if:

1. $b_n$ is decreasing eventually, and
2. ${\lim }_{n\to \infty }{b}_n=0$.

This is a standard BC result and often faster than ratio or integral tests.

Absolute vs conditional convergence

For alternating series, also ask whether $$\sum |a_n|$$ converges.

• If $\sum |a_n|$ converges, the series is absolutely convergent (stronger).
• If $\sum a_n$ converges but $\sum |a_n|$ diverges, it is conditionally convergent (common with alternating p-series where $p\le 1$ is not allowed for absolute convergence).

Step 4: Use Ratio or Root Tests for factorials, exponentials, and powers

When terms involve $n!$, $a^n$, or complicated products, the Ratio Test is often the clearest.

Ratio Test

Compute $$L=\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|.$$

• If $L<1$, the series converges absolutely.
• If $L>1$ (or $=\infty$), the series diverges.
• If $L=1$, the test is inconclusive.

Root Test

Compute $$L=\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}.$$ Same conclusions as the Ratio Test. Root is often clean when the whole term is raised to the $n$th power.

Step 5: Use the Integral Test when terms look like $f(n)$

If $a_n=f(n)$ where $f$ is continuous, positive, decreasing, and integrable on $[1,\infty )$, then $$\sum a_n\text{ and }{\int }_1^{\infty }f(x)dx$$ either both converge or both diverge.

This is a natural choice for series like $\sum \frac{1}{n(\ln n{)}^p}$ (a classic comparison and integral-test family), though you still need to verify the conditions.

Power Series: Convergence as an Interval, Not a Yes/No

A power series has the form $$\sum _{n=0}^{\infty }{c}_n(x-a{)}^n.$$

Unlike numeric series, power series converge for some $x$ and diverge for others. The key goal is to find the radius of convergence $R$ and then the interval of convergence.

Radius of convergence (often via the Ratio Test)

Applying the Ratio Test typically produces an inequality like $|x-a|<R$. The set of $x$ values that satisfy this is the open interval $(a-R,a+R)$.

After finding $R$, always test endpoints separately:

• Plug in $x=a-R$ and analyze the resulting numeric series.
• Plug in $x=a+R$ and analyze again.

Endpoints can behave differently, so the interval of convergence might include neither, one, or both endpoints.

Power series behave nicely inside their interval

A central BC fact: within the interval of convergence, you can differentiate and integrate a power series term-by-term, and the new series shares the same radius of convergence.

If $$\sum c_n(x-a{)}^n$$ converges for $|x-a|<R$, then $$\frac{d}{dx}\left(\sum c_n(x-a{)}^n\right)=\sum n{c}_n(x-a{)}^{n-1}$$ and $$\int \left(\sum c_n(x-a{)}^n\right)dx=C+\sum \frac{c_n}{n+1}(x-a{)}^{n+1}.$$

These operations are the engine behind many exam problems that ask for new series from known ones.

Taylor and Maclaurin Series: Building Functions from Derivatives

A Taylor series for a function $f$ centered at $x=a$ is $$\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n.$$

A Maclaurin series is just a Taylor series centered at $a=0$: $$\sum _{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n.$$

Why these series matter in BC

Taylor and Maclaurin series provide:

• Polynomial approximations near the center
• Tools for evaluating limits
• Series methods for integration and differential equations (at the BC level, typically integration and approximation)
• A structured way to generate new series from known ones using substitution, differentiation, and integration

Using known Maclaurin series as templates

You are expected to work fluently with standard expansions and transform them. Typical moves include:

• Replace $x$ with $kx$ or $(x-a)$
• Multiply by a polynomial factor
• Differentiate or integrate to create a related series
• Combine series through addition or subtraction

For example, if you know a base series for a function, then $f(2x)$ typically changes the radius of convergence by a factor of $\frac{1}{2}$ because $|2x|<R$ becomes $