AP Calculus BC Unit 10: Convergence Tests for Infinite Series
Integral Test for Convergence
A big challenge with infinite series is that you can’t literally “add infinitely many terms” by hand. Convergence tests give you strategies to decide whether a series has a finite sum (converges) or grows without bound or fails to settle (diverges). The Integral Test is one of the most conceptually satisfying tests because it connects series to something you already know well from calculus: improper integrals.
What the Integral Test is
Suppose you have a series
\sum_{n=1}^{\infty} a_n
The Integral Test applies when the terms come from a function. Specifically, you look for a function f such that
a_n = f(n)
and f is:
- positive for x \ge 1
- continuous for x \ge 1
- decreasing for x \ge 1
If those conditions hold, then the Integral Test says:
\sum_{n=1}^{\infty} a_n
and
\int_{1}^{\infty} f(x)\,dx
either both converge or both diverge.
Why it matters
The Integral Test matters because many series look like “discretized versions” of areas under curves. When f(x) is decreasing and positive, the rectangles of width 1 and height f(n) approximate the area under f. If the total area under the curve from 1 to infinity is finite, then the “total rectangle area” (the series) is also finite, and vice versa.
This test is especially useful for series that resemble
\sum \frac{1}{n^p}
or involve logarithms, where comparisons can be tricky but integrals are manageable.
How it works (the idea)
For a positive decreasing function, the integral and the series squeeze each other. Geometrically, using left-endpoint or right-endpoint rectangles gives inequalities that relate partial sums to integrals. You don’t usually need to reproduce the proof on the AP exam, but you do need to know the conditions and be able to evaluate an improper integral.
Notation you’ll see (and what it means)
| Object | Common notation | Meaning |
|---|---|---|
| Infinite series | \sum_{n=1}^{\infty} a_n | The “infinite sum” you’re testing |
| Partial sum | S_N = \sum_{n=1}^{N} a_n | Finite sum up to term N |
| Series sum (if it exists) | S = \sum_{n=1}^{\infty} a_n | Limit of partial sums, if it converges |
| Remainder (error) after N terms | R_N = S - S_N | How far the partial sum is from the true sum |
Example 1: A classic p-series via Integral Test
Determine whether
\sum_{n=1}^{\infty} \frac{1}{n^2}
converges.
Step 1: Match to a function. Let
f(x) = \frac{1}{x^2}
For x \ge 1, f is positive, continuous, and decreasing.
Step 2: Evaluate the improper integral.
\int_{1}^{\infty} \frac{1}{x^2}\,dx = \lim_{b\to\infty} \int_{1}^{b} x^{-2}\,dx
Antiderivative:
\int x^{-2}\,dx = -x^{-1} + C
So
\lim_{b\to\infty} \left[-\frac{1}{x}\right]_{1}^{b} = \lim_{b\to\infty} \left(-\frac{1}{b} + 1\right) = 1
The integral converges, so the series converges.
Example 2: A series that diverges (harmonic-type)
Test
\sum_{n=1}^{\infty} \frac{1}{n}
Let
f(x) = \frac{1}{x}
Then
\int_{1}^{\infty} \frac{1}{x}\,dx = \lim_{b\to\infty} \int_{1}^{b} \frac{1}{x}\,dx = \lim_{b\to\infty} \left[\ln x\right]_{1}^{b} = \lim_{b\to\infty} \ln b
This diverges, so the series diverges.
What goes wrong (common pitfalls)
A frequent mistake is using the Integral Test when f is not decreasing or not positive. The test is not “integral equals series”; it’s only a convergence/divergence link under specific conditions. Another common error is evaluating the improper integral incorrectly—especially forgetting the limit as the upper bound goes to infinity.
Exam Focus
- Typical question patterns:
- “Use the Integral Test to determine convergence/divergence” (you must state conditions and compute the improper integral).
- “Explain why the Integral Test applies” (justify positivity, continuity, decreasing behavior).
- Occasionally: interpret convergence in terms of finite area under a curve.
- Common mistakes:
- Forgetting to verify f is decreasing on [1,\infty).
- Computing \int_{1}^{\infty} f(x)\,dx but not explicitly concluding what it implies for the series.
- Algebra errors with logs or power antiderivatives in improper integrals.
Comparison Tests
Comparison tests are your main tools when a series looks similar to a series you already understand. The big idea is: if you can trap a complicated series above or below by a simpler benchmark series, you can “inherit” convergence or divergence.
Why comparisons are so powerful
In Unit 10, you build a library of known behaviors. Two especially important benchmarks are:
- Geometric series
\sum_{n=0}^{\infty} ar^n
(converges if |r|