10.1 Defining Convergent and Divergent Infinite Series

TLDR

An infinite series converges when its sequence of partial sums approaches a finite limit, and it diverges when that limit does not exist or is infinite. In AP Calculus BC, you find the sum of a series by taking the limit of its partial sums, written as $S = \lim_{n \to \infty} S_n$.

What Makes an Infinite Series Converge or Diverge?

An infinite series $\sum_{n=1}^{\infty} a_n$ converges when the sequence of partial sums $S_n=\sum_{i=1}^{n} a_i$ approaches a finite real number. If $\lim_{n \to \infty} S_n$ does not exist or is infinite, the series diverges.

For AP Calculus BC, keep the key distinction clear: a sequence asks what happens to the terms $a_n$, while a series asks what happens to the running totals $S_n$. Later convergence tests are shortcuts for deciding whether those partial sums settle on a finite value.

Why This Matters for the AP Calculus Exam

This is the first topic in the BC-only series unit, and it sets up everything that follows. Once you understand that a series converges only when its partial sums settle on a finite value, the convergence tests in later topics make more sense, since each one is just a shortcut for checking that behavior.

On the AP Calculus BC exam, you will see infinite series in both multiple-choice and free-response questions. This foundational topic shows up when you need to:

• State what convergence means using the limit of partial sums.
• Recognize telescoping series, where most terms cancel and you can find an exact sum.
• Set up the partial sum $S_n$ before applying a test or finding a limit.

Clear notation here matters for the rest of the unit, since later tests build directly on the partial-sum definition.

Key Takeaways

• A series is the sum of the terms of a sequence, written $\sum_{n=1}^{\infty} a_n$.
• The nth partial sum $S_n$ is the sum of just the first $n$ terms: $S_n = \sum_{i=1}^{n} a_i$.
• A series converges to $S$ exactly when $\lim_{n \to \infty} S_n = S$ exists and is finite.
• A series diverges when $\lim_{n \to \infty} S_n$ does not exist or is infinite.
• For a telescoping series, most middle terms cancel, so you can simplify $S_n$ and take its limit directly.
• Convergence of a series depends on the limit of the partial sums, not on whether the individual terms get small.

What is a Sequence?

A sequence is a list of terms related by a common pattern. You write a sequence like this:

$$\{a_n\}^\infty_1$$

A series is built from a sequence, so getting comfortable with sequence notation first makes the rest easier.

Finding Terms in a Sequence: Example 1

List $a_1, a_2, a_3, a_n$, and $a_{n+1}$ for

$$\{\dfrac1n\}^\infty_1$$

Plug in $n = 1, 2, 3, n,$ and $n+1$ into $a_n$:

$$\{1, \dfrac12, \dfrac13,...,\frac{1}{n}, \frac{1}{n+1}, ...\}$$

The ... between the 3rd and $n^{th}$ term and after the $(n+1)^{th}$ term shows there are infinitely many terms in those gaps.

This is the harmonic sequence. You will see more about it in 10.5 Harmonic Series and p-Series.

Finding Terms in a Sequence: Example 2

$$\{\frac{(-1)^n \cdot n!}{2^n}\}^\infty_1$$

Same process as before, but you can use algebra to rewrite $a_{n+1}$:

$$a_1 = -\dfrac12$$ $$a_2 = \dfrac12$$ $$a_n = \frac{(-1)^n \cdot n!}{2^n}$$ $$a_{n+1} = \frac{(-1)^{n+1}\cdot(n+1)!}{2^{n+1}} = -\frac{(-1)^n(n+1)n!}{2(2^n)} = -a_n\cdot\frac{n+1}{2}$$

The $(-1)^n$ tells you this is an alternating sequence. You will see more about these in 10.7 Alternating Series Test for Convergence.

The Limit of a Sequence

Like functions, sequences have limits. You find them much the same way, but in this unit you only care about the limit as $n$ approaches $\infty$. All limit properties that hold for functions also hold for sequences.

The first thing this lets you check is whether a sequence is convergent or divergent. These two words show up constantly in this unit.

1. Convergent Sequence: A sequence where $\lim\limits_{n \to \infty} a_n$ exists and is finite.
2. Divergent Sequence: A sequence where $\lim\limits_{n \to \infty} a_n$ does not exist or is infinite.

Convergence and Divergence of Sequences

Determine whether each sequence converges or diverges.

Sequences Example 1

$$\{\frac{(-1)^n}{n}\}^\infty_1$$ $$= \lim\limits_{n \to \infty} \frac{(-1)^n}{n}$$ $$= \lim\limits_{n \to \infty} (-1)^n\cdot\dfrac1n = 0$$

The limit is finite, so the sequence converges.

Sequences Example 2

$$\{\frac{n^2+1}{n}\}^\infty_1$$

Using what you know about limits of rational functions, $= \lim\limits_{n \to \infty} \frac{\ n^2 \ +1}{n} = \infty$

So this sequence diverges.

Sequences Example 3

$$\{(-1)^n\}^\infty_1$$

Treat this sequence as a piecewise pattern:

$$a_n=\begin{cases}1& n\%2=0\\ -1& n\%2=1\\\end{cases}$$

As $n$ approaches infinity, the terms keep bouncing between 1 and -1, so the limit does not exist and the sequence diverges.

Notation and Terminology on Sequences

Before moving to series, here are a few sequence terms worth knowing.

• Increasing Sequence: $\frac{a_{n+1}}{a_n} > 1$ for all $n$, so each term is greater than the last.
• Decreasing Sequence: $\frac{a_{n+1}}{a_n} < 1$ for all $n$, so each term is less than the last.
• Monotonic Sequence: A sequence that is either increasing or decreasing.
• Bounded Above: There is an upper bound the terms do not go above.
• Bounded Below: There is a lower bound the terms do not go below.
• Bounded: The sequence is bounded both above and below.

Sequence Convergence Theorem

If a sequence is both bounded and monotonic, then the sequence converges. Note: failing to meet both conditions does not automatically mean it diverges.

What is a Series?

A series is the sum of the terms in a sequence. You can write it like this:

$$s_n = \sum_{i=1}^{n} a_i$$

where $s_n$ adds up all the terms from the 1st through the $n^{th}$, inclusive. This $s_n$ is the nth partial sum. An infinite series keeps going forever:

$$s_\infty = \lim\limits_{n \to \infty} \sum_{i=1}^{n} a_i$$

You usually cannot find an exact sum for an infinite series, but the next example is a case where you can.

Finding Partial Sums

Find the partial sums $s_1, s_2, s_3, s_n$, and $s_\infty$ of the series with $a_n=\frac{1}{n^2+n}$.

First find $a_1, a_2,$ and $a_3$:

$a_1=\frac {1}{2}$

$a_2=\frac {1}{6}$

$a_3=\frac {1}{12}$

Now build the first three partial sums:

$$s_1 = a_1 = \dfrac12$$ $$s_2 = a_1 + a_2 = \dfrac46 = \dfrac23$$ $$s_3 = a_1+a_2+a_3 = \dfrac{9}{12} = \dfrac34$$

To find $s_n$ and $s_\infty$, do a partial fraction decomposition of the term formula. You likely will not have to do this on the exam, but it shows that $s_\infty$ can sometimes be found exactly.

$$a_n = \frac{1}{n^2+n} = \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1}$$

Now find $s_n$ and $s_\infty$:

$$s_n = \frac{1}{1} - \dfrac12 + \dfrac12 - \dfrac13 + \dfrac13 +...+ \dfrac{1}{n-1} - \dfrac{1}{n} + \dfrac{1}{n} - \dfrac{1}{n+1}$$ $$= 1 - \frac{1}{n+1}$$

The middle terms cancel, leaving only the first and last. A series that behaves this way is a telescoping series, like a collapsible telescope where the middle collapses and the ends remain. Now find $s_\infty$:

$$s_\infty = \lim\limits_{n \to \infty} s_n$$ $$= \lim\limits_{n \to \infty} 1-\frac{1}{n+1}$$ $$=1$$

Convergence and Divergence of Series

Like sequences, series can converge or diverge.

1. Convergent Series: A series where $s_\infty$ exists and is finite.
2. Divergent Series: A series where $s_\infty$ does not exist or is infinite.

Since the series above has a finite infinite partial sum, it converges. Through the rest of this part of the unit you will learn shortcuts to decide convergence or divergence, so do not worry if it is not obvious yet.

A few helpful properties of convergent series: if $s_n$ and $t_n$ are convergent series and $c$ is a constant, then

1. The series $r_n = c\cdot s_n$ converges.
2. The series $q_n = s_n \pm t_n$ converges.

How to Use This on the AP Calculus Exam

Problem Solving

• Read the term formula $a_n$ carefully, then build $S_1, S_2, S_3$ to see the pattern before jumping to a limit.
• For a telescoping series, write out enough terms to see exactly which ones cancel, then simplify $S_n$ to a short expression before taking the limit.
• To decide convergence, evaluate $\lim_{n \to \infty} S_n$. A finite value means convergence; an infinite or nonexistent limit means divergence.

Common Trap

• Convergence is about the limit of the partial sums $S_n$, not the limit of the individual terms $a_n$. Those are different questions, and confusing them leads to wrong conclusions later in the unit.

Partial Sums: Practice Problems

For each sequence $a_n$, find $a_{n+1}$ in terms of $a_n$ if possible, then solve for the partial sums $s_1, s_2,$ and $s_3$.

$$1. \ a_n = \frac{1}{e^n}$$ $$2.\ a_n = \frac{n}{(n+2)!}$$

Solution to Practice Problem 1

$$a_n = \frac{1}{e^n}$$ $$a_{n+1}= e^{-(n+1)}$$ $$s_1=e^{-1}$$ $$s_2=e^{-1}+e^{-2}$$ $$s_3=e^{-1}+e^{-2}+e^{-3}$$

Solution to Practice Problem 2

$$a_n = \frac{n}{(n+2)!}$$ $$a_{n+1}=\frac{(n+1)}{(n+3)!}$$ $$s_1=\frac{1}{6}$$ $$s_2=\frac{1}{4}$$ $$s_3=\frac{11}{40}$$

Common Misconceptions

• "If the terms $a_n$ go to 0, the series converges." Not true. The terms going to 0 is necessary but not enough. The harmonic series $\sum \frac{1}{n}$ has terms going to 0 yet diverges.
• "Convergence of a sequence and convergence of a series are the same thing." A sequence converges if its terms approach a finite limit. A series converges if its partial sums approach a finite limit. You can have a sequence whose terms converge to 0 while the series still diverges.
• "A divergent series just means it goes to infinity." Divergence also covers cases where the partial sums oscillate and never settle, like $\sum (-1)^n$. Divergence simply means the limit of partial sums does not exist as a finite number.
• "Every infinite series has a findable sum." Most do not give a clean closed form. Telescoping and geometric series are special cases where you can compute the exact sum.
• "Bounded means convergent." A sequence must be both bounded and monotonic for that theorem to guarantee convergence. Bounded alone is not enough.

Related AP Calculus Guides

• Unit 10 Overview: Infinite Series and Sequences
• 10.3 The nth Term Test for Divergence
• 10.5 Harmonic Series and p-Series
• 10.2 Working with Geometric Series
• 10.4 Integral Test for Convergence
• 10.6 Comparison Tests for Convergence