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Infinite Series:
Convergence & Divergence

Learn how to determine whether an infinite series settles to a finite sum or grows without bound: the foundation of all series analysis in calculus.

AP Calculus BC

What Is an Infinite Series?

An infinite series is the sum of infinitely many terms from a sequence \( \{a_n\} \):

The big question: can you add infinitely many numbers and get a finite result? Sometimes yes, sometimes no. That distinction is the heart of this lesson.

The Partial Sum Sequence

We can't add infinitely many numbers directly. Instead, we build a sequence of partial sums:

Each \( S_N \) is a finite sum. As \( N \to \infty \), the behavior of the sequence \( \{S_N\} \) tells us whether the infinite series converges or diverges.

Visualizing Partial Sums

Consider \( \displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n} \). The partial sums approach a limit:

S = 1 0 0.5 1.0 N (number of terms) 0.5 0.75 0.875 0.9375 1 2 3 4 5 6 7

Formal Definitions

Convergent
If \( \displaystyle\lim_{N\to\infty} S_N = S \) exists and is finite, the series converges and its sum is \( S \).
Divergent
If \( \displaystyle\lim_{N\to\infty} S_N \) does not exist or is infinite, the series diverges.

The Geometric Series Theorem

Converges
When \( |r| < 1 \), the terms shrink fast enough for the sum to be finite.
Diverges
When \( |r| \geq 1 \), the terms don't shrink (or grow), so the sum blows up.

This is the most important convergent series in calculus: it gives an exact closed-form sum.

A bouncing ball leaves a shrinking ladder of travel

An orange basketball rises above a gym floor while two faint silhouettes suggest earlier and later stages of diminishing rebounds; their heights are illustrative.A ball falls 10 m and rebounds to 60% of its preceding height. Count the initial fall once and each rebound twice: the geometric series gives 40 m of total travel.

A bouncing ball leaves a shrinking ladder of travel

A ball falls 10 m and rebounds to 60% of its preceding height. Count the initial fall once and each rebound twice: the geometric series gives 40 m of total travel.A ball falls 10 m and rebounds to 60% of its preceding height. Count the initial fall once and each rebound twice: the geometric series gives 40 m of total travel.

A Famous Divergent Series

The harmonic series \( \displaystyle\sum_{n=1}^{\infty} \frac{1}{n} \) diverges, even though \( a_n \to 0 \):

0 1 2 3 4 N (number of terms) ∞ 1 3 5 7 9

The nth-Term Test for Divergence

This is the simplest divergence test: if the terms don't approach zero, the series cannot converge. But be careful of the converse:

What it says
\( a_n \not\to 0 \implies \) diverges
What it doesn't say
\( a_n \to 0 \) does NOT guarantee convergence (harmonic series!)

Sequence vs Series

Sequence \(\{a_n\}\) Series \(\sum a_n\)
What it is An ordered list of numbers The sum of those numbers
Convergence means \(a_n \to L\) (single number) \(S_N \to S\) (partial sums)
Example \(\frac{1}{n} \to 0\) ✓ \(\sum \frac{1}{n} = \infty\) ✕
Key insight Sequence converging does not force the series to converge

Telescoping Series

A telescoping series is one where most terms in the partial sum cancel, making it easy to evaluate the limit:

After cancellation, \( S_N = 1 - \frac{1}{N+1} \). Since \( S_N \to 1 \), the series converges to 1.

How to Determine Convergence

1
nth-Term Test: Compute \( \lim_{n\to\infty} a_n \). If it's not 0, the series diverges. Stop.
2
Recognize the type: Is it geometric (\( ar^n \))? Telescoping? A known series like \( p \)-series?
3
Write \( S_N \): If possible, find a closed form for the partial sum and take its limit.
4
Evaluate the limit: If \( \lim_{N\to\infty} S_N = S \) (finite), it converges to \( S \). Otherwise, it diverges.

Example 1: Geometric Series

Determine whether \( \displaystyle\sum_{n=0}^{\infty} 3\!\left(\frac{2}{5}\right)^n \) converges. If so, find its sum.

1
Identify: This is geometric with \( a = 3 \) and \( r = \frac{2}{5} \).
2
Check \( |r| \): \( \left|\frac{2}{5}\right| = 0.4 < 1 \), so the series converges.
3
Compute the sum: \( \displaystyle S = \frac{a}{1-r} = \frac{3}{1 - \frac{2}{5}} = \frac{3}{\frac{3}{5}} = 5 \).

Example 2: Telescoping Series

Find \( \displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+1)} \).

1
Partial fractions: \( \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} \).
2
Write \( S_N \): \( \left(1 - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \cdots = 1 - \frac{1}{N+1} \).
3
Take the limit: \( \displaystyle\lim_{N\to\infty}\left(1 - \frac{1}{N+1}\right) = 1 \).

Example 3: Divergence by nth-Term Test

Does \( \displaystyle\sum_{n=1}^{\infty} \frac{n}{2n+1} \) converge or diverge?

1
Apply the nth-Term Test: Compute \( \displaystyle\lim_{n\to\infty} \frac{n}{2n+1} \).
2
Divide top and bottom by \( n \): \( \displaystyle\lim_{n\to\infty} \frac{1}{2 + \frac{1}{n}} = \frac{1}{2} \neq 0 \).
3
Conclude: Since \( a_n \to \frac{1}{2} \neq 0 \), the series diverges.

Common Mistakes

✕ Converse of nth-Term Test
Claiming \( a_n \to 0 \) means the series converges. Counterexample: \( \sum \frac{1}{n} \) diverges even though \( \frac{1}{n} \to 0 \).
✕ Confusing Sequence and Series
Saying "\( \{a_n\} \) converges" when you mean "\( \sum a_n \) converges." These are different statements with different criteria.
✕ Wrong Geometric Formula
Using \( \frac{a}{1-r} \) when the series starts at \( n=1 \) instead of \( n=0 \). First term matters: adjust \( a \) accordingly.

Series Summary Table

Series Converges? Sum / Behavior
\( \sum ar^n \), \( |r| < 1 \) Yes \( \frac{a}{1-r} \)
\( \sum ar^n \), \( |r| \geq 1 \) No Diverges
\( \sum \frac{1}{n} \) (harmonic) No \( S_N \to \infty \)
\( \sum \frac{1}{n(n+1)} \) (telescoping) Yes 1
\( a_n \not\to 0 \) (nth-term test) No Diverges (always)

Question 1 of 5

What is the sum of the geometric series \( \displaystyle\sum_{n=0}^{\infty} 4\!\left(\frac{1}{3}\right)^n \)?

Question 2 of 5

Which of the following is true about the harmonic series \( \displaystyle\sum_{n=1}^{\infty} \frac{1}{n} \)?

Question 3 of 5

If \( \displaystyle\lim_{n\to\infty} a_n = 5 \), what can you conclude about \( \displaystyle\sum_{n=1}^{\infty} a_n \)?

Question 4 of 5

Find the sum of the telescoping series \( \displaystyle\sum_{n=1}^{\infty} \left(\frac{1}{n+1} - \frac{1}{n+2}\right) \).

Question 5 of 5

Which series converges?