Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read · Updated 2026-05-10
Integral Test for Convergence — AP Calculus BC
AP Calculus BC · Unit 10: Infinite Sequences and Series · 14 min read
1. The Integral Test and Required Hypotheses★★☆☆☆⏱ 4 min
The integral test for convergence connects the behavior of an infinite discrete positive-term series to the convergence of an improper integral, leveraging your existing knowledge of integration. For a series $\sum_{n=N}^\infty a_n$ where $a_n = f(n)$, convergence of the series matches convergence of the improper integral $\int_N^\infty f(x) dx$ when all three hypotheses are satisfied.
$f$ is **continuous** on $[N, \infty)$: no discontinuities, jumps, or vertical asymptotes in the interval
$f$ is **positive** on $[N, \infty)$: all outputs of $f$ for $x \geq N$ are greater than 0
$f$ is **decreasing** on $[N, \infty)$: $f'(x) \leq 0$ for all $x > N$, so $f(x)$ does not increase as $x$ increases
2. Applying the Integral Test for Convergence★★☆☆☆⏱ 4 min
Once all three hypotheses are verified, the integral test gives a clear conclusion: the series $\sum_{n=N}^\infty a_n$ converges if and only if the improper integral $\int_N^\infty f(x) dx$ converges. A finite integral means the series converges; a divergent integral means the series diverges.
This relationship comes from Riemann sum bounding for decreasing positive functions:
The most famous result from the integral test is the p-series convergence rule: $\sum_{n=1}^\infty \frac{1}{n^p}$ converges if $p>1$ and diverges if $p \leq 1$, which is proven directly using the integral test.
3. Remainder Error Bounds for Partial Sums★★★☆☆⏱ 3 min
When you approximate the sum of a convergent infinite series with its $n$-th partial sum $S_n$, the remainder $R_n = S - S_n$ is the error in your approximation. For series satisfying the integral test hypotheses, we can find explicit upper and lower bounds for this error.
If $f$ is continuous, positive, and decreasing for $x \geq n$, and $\sum_{k=1}^\infty f(k) = S$ converges, the remainder satisfies:
4. AP-Style Concept Check★★★☆☆⏱ 3 min
Common Pitfalls
Why: The positivity condition is violated, since $\frac{\sin x}{x^2}$ alternates sign.
Why: Students confuse definite integrals with improper integrals, forgetting we need the integral to infinity to apply the test.
Why: Students confuse the value of the integral with the sum of the series.
Why: Students mix up the nth term test with the integral test, incorrectly attributing the divergence conclusion to the wrong test.
Why: Students memorize the formula without remembering it only applies to convergent series, since divergent series have no finite sum to approximate.