Integral Test for Convergence

AP Course

AP Calculus AB/BC

Study the complete College Board sequence for AP Calculus AB and BC, from limits through infinite series.

Choose an official unit and topic to open its lecture, concept check, or focused practice.

AP Calculus AB/BC · Unit 10 · Topic 10.4 · BC Only

Integral Test for Convergence

Relate a positive series to an improper integral of a continuous decreasing extension and use the same geometry to bound its remainder.

1. Topic Focus

Determine series convergence, estimate error, construct Taylor approximations, and represent functions with power series on valid intervals.

This topic: Relate a positive series to an improper integral of a continuous decreasing extension and use the same geometry to bound its remainder.

2. Key Relationship

\(\sum_{n=N}^{\infty}f(n)\text{ and }\int_N^\infty f(x)\,dx\text{ share convergence}\)

Read every symbol with its domain, direction, units, and hypotheses before applying the relationship.

3. Visual Connection

Convergence comparisonVerify each test's hypotheses, then compare decay with an integral, p-series, ratio, or positive benchmark.

4. Worked Example

Verify positivity, continuity, and eventual decrease before comparing rectangles with the area under the curve.

Write the governing relationship first, carry out the algebra cleanly, and finish with a sentence that answers the mathematical question.

5. Concept Development

BC-only topic. The Integral Test converts a question about infinitely many discrete terms into a question about the area under a related continuous curve. It determines convergence, and its rectangle comparison also yields useful error bounds.

Integral Test

Suppose \(a_n=f(n)\), and for all \(x\ge N_0\), the function \(f\) is positive, continuous, and decreasing. Then

\(\boxed{\sum_{n=N_0}^{\infty}a_n\text{ converges}\quad\Longleftrightarrow\quad\int_{N_0}^{\infty}f(x)\,dx\text{ converges}.}\)

If the improper integral diverges, the series diverges; if the integral converges, the series converges.

Why the test works

For a positive decreasing function, unit-width rectangles with heights \(f(n)\) can be compared with the area under \(y=f(x)\). The rectangles and the integral differ only by bounded edge pieces. Therefore one accumulation is finite exactly when the other is finite.

\(\int_{N_0}^{M+1}f(x)dx\le\sum_{n=N_0}^{M}f(n)\le f(N_0)+\int_{N_0}^{M}f(x)dx.\)

Verify every hypothesis

Positive: \(f(x)>0\) on the relevant tail.
Continuous: no undefined points or vertical asymptotes occur on that tail.
Decreasing: show \(f'(x)\le0\), compare values, or give another valid argument.
Matches the terms: confirm \(f(n)=a_n\) at integer inputs.

The conditions need only hold eventually. A finite number of exceptional starting terms cannot change convergence.

Evaluate the improper integral correctly

Replace infinity by a limit:

\(\int_{N_0}^{\infty}f(x)dx=\lim\limits_{b\to\infty}\int_{N_0}^{b}f(x)dx.\)

A finite limit means the integral converges. An infinite or nonexistent limit means it diverges.

The integral is not the series sum

The Integral Test establishes shared convergence behavior, not equality:

\(\sum_{n=N_0}^{\infty}f(n)\ne\int_{N_0}^{\infty}f(x)dx\quad\text{in general}.\)

The integral represents curved area; the series represents rectangle heights. Their difference is precisely what later bounds exploit.

Standard antiderivative patterns

\(\int x^{-p}dx\) connects to \(p\)-series.
\(\int dx/[x(\ln x)^p]\) uses \(u=\ln x\).
\(\int dx/(1+x^2)=\arctan x+C\).
Exponential decay such as \(e^{-x}\) usually gives a finite improper integral.

The logarithmic boundary family

For \(n\ge2\), consider

\(\sum\frac1{n(\ln n)^p}.\)

With \(u=\ln x\),

\(\int_2^\infty\frac{dx}{x(\ln x)^p}=\int_{\ln2}^{\infty}u^{-p}du.\)

Thus the series converges when \(p>1\) and diverges when \(p\le1\).

When the Integral Test is a poor choice

Do not force this test onto alternating or sign-changing terms, a discontinuous extension, or a function whose improper integral is harder than the original series. Another test may be more natural even if some continuous extension exists.

Remainder estimate

Suppose the positive series converges and \(f\) satisfies the Integral Test hypotheses. If

\(S_N=\sum_{n=N_0}^{N}a_n,\qquad R_N=S-S_N=\sum_{n=N+1}^{\infty}a_n,\)

then the decreasing-rectangle geometry gives

\(\boxed{\int_{N+1}^{\infty}f(x)dx\le R_N\le\int_N^\infty f(x)dx.}\)

Bounds for the unknown sum

Add \(S_N\) to the remainder bounds:

\(S_N+\int_{N+1}^{\infty}f(x)dx\le S\le S_N+\int_N^\infty f(x)dx.\)

This produces a guaranteed interval for the series sum without knowing its exact value.

Choose enough terms for a tolerance

To guarantee \(|R_N|<\varepsilon\), it is sufficient to solve

\(\int_N^\infty f(x)dx<\varepsilon\)

and round \(N\) upward as required. Check strict versus non-strict inequalities at the final integer.

Integral-Test checklist

Perform the nth term test first.
Define a natural continuous function \(f(x)\) with \(f(n)=a_n\).
Verify positivity, continuity, and decrease on an appropriate tail.
Write the improper integral with a finite upper limit \(b\).
Evaluate its limit and state the matching series conclusion.
If estimating, distinguish \(S_N\), \(R_N\), the integral bounds, and the true sum \(S\).

6. Detailed Worked Example and Error Check

Example 1: A convergent logarithmic series

For \(n\ge2\), let \(f(x)=1/[x(\ln x)^2]\). It is positive and continuous, and

\(f'(x)=-\frac{\ln x+2}{x^2(\ln x)^3}<0\quad(x\ge2).\)

Using \(u=\ln x\),

\(\int_2^\infty\frac{dx}{x(\ln x)^2}=\left[-\frac1{\ln x}\right]_2^\infty=\frac1{\ln2}.\)

The integral converges, so the series converges. The value \(1/\ln2\) is the integral value, not the series sum.

Example 2: The logarithmic boundary diverges

For \(f(x)=1/(x\ln x)\), positive, continuous, and decreasing for \(x\ge2\),

\(\int_2^\infty\frac{dx}{x\ln x}=\lim\limits_{b\to\infty}[\ln(\ln x)]_2^b=\infty.\)

Therefore \(\sum_{n=2}^{\infty}1/(n\ln n)\) diverges.

Example 3: A convergent power

For \(f(x)=x^{-3/2}\),

\(\int_1^\infty x^{-3/2}dx=\lim\limits_{b\to\infty}\left[-2x^{-1/2}\right]_1^b=2.\)

The series \(\sum1/n^{3/2}\) converges. Its sum is not asserted to equal \(2\).

Example 4: A divergent power

For \(f(x)=x^{-1/2}\),

\(\int_1^\infty x^{-1/2}dx=\lim\limits_{b\to\infty}2(\sqrt b-1)=\infty.\)

Thus \(\sum1/\sqrt n\) diverges.

Example 5: Classify an entire logarithmic family

For \(\sum_{n=2}^{\infty}1/[n(\ln n)^p]\), substitute \(u=\ln x\). The corresponding integral becomes \(\int_{\ln2}^{\infty}u^{-p}du\). Therefore the series converges for \(p>1\) and diverges for \(p\le1\).

Example 6: An inverse-trigonometric integral

Let \(a_n=1/(n^2+1)\). The extension \(f(x)=1/(x^2+1)\) is positive, continuous, and decreasing for \(x\ge1\):

\(\int_1^\infty\frac{dx}{x^2+1}=\left[\arctan x\right]_1^\infty=\frac{\pi}{4}.\)

The series converges.

Example 7: Exponential decay

For \(a_n=e^{-n}\), use \(f(x)=e^{-x}\):

\(\int_1^\infty e^{-x}dx=e^{-1}.\)

The Integral Test proves convergence, although recognizing the series as geometric is faster.

Example 8: Eventually decreasing is enough

For \(a_n=n/e^n\), take \(f(x)=xe^{-x}\). Since

\(f'(x)=e^{-x}(1-x)<0\quad(x>1),\)

the function is decreasing on the relevant tail, and

\(\int_1^\infty xe^{-x}dx=\frac2e.\)

Thus \(\sum n/e^n\) converges.

Example 9: Bound a remainder

For \(\sum1/n^3\), after \(N=10\),

\(\frac1{2(11)^2}=\int_{11}^{\infty}x^{-3}dx\le R_{10}\le\int_{10}^{\infty}x^{-3}dx=\frac1{200}.\)

Therefore \(S_{10}\) underestimates the sum by at most \(0.005\).

Example 10: Choose \(N\) for a required error

To approximate \(\sum1/n^3\) with error less than \(0.001\), require

\(\frac1{2N^2}<0.001\quad\Longrightarrow\quad N>\sqrt{500}\approx22.36.\)

The least integer guaranteed by this upper bound is \(N=23\).

Common errors

Failing to verify positivity, continuity, or decrease.
Claiming the series equals the improper integral.
Evaluating an improper integral without writing a limit.
Using a divergent antiderivative expression as though it were a finite number.
Applying the test to sign-changing terms without modification or justification.
Requiring conditions from the first term when they hold on a later tail.
Forgetting the substitution \(u=\ln x\) in logarithmic families.
Reversing the \(N\) and \(N+1\) remainder bounds.
Using an integral bound for a series not yet shown to converge.
Rounding \(N\) down when an error guarantee requires rounding up.

7. AP Reasoning Routine

Check the nth-term condition first, match the series structure to a justified test, state convergence type, and test power-series endpoints separately.

Identify the representation and requested quantity.
State the rule or theorem and verify its conditions.
Keep exact values until the final requested approximation.
Interpret sign, units, interval, and context.

AP Checkpoint

Use the Integral Test or its remainder estimate, verifying all required conditions.
(a) Determine the convergence of \(\sum_{n=2}^{\infty}1/[n(\ln n)^3]\).
(b) Determine the convergence of \(\sum_{n=2}^{\infty}1/(n\ln n)\).
(c) Determine the convergence of \(\sum_{n=1}^{\infty}1/(n+1)^2\).
(d) Determine the convergence of \(\sum_{n=2}^{\infty}n/(n^2+1)\).
(e) Determine the convergence of \(\sum_{n=1}^{\infty}e^{-\sqrt n}\).
(f) Determine the convergence of \(\sum_{n=3}^{\infty}1/[n\ln n\ln(\ln n)]\).
(g) Explain why \(f(x)=2+\sin x\) does not satisfy the Integral Test hypotheses for \(\sum(2+\sin n)\), and classify the series another way.
(h) For \(\sum1/n^2\), give upper and lower integral bounds for \(R_N\).
(i) Find the least integer \(N\) guaranteed by the integral upper bound to make the error in \(S_N\) for \(\sum1/n^2\) less than \(0.01\).
(j) Compute \(S_5\) for \(\sum1/n^2\) and give an interval guaranteed to contain the total sum.
(k) Explain why \(\int_1^\infty x^{-2}dx=1\) does not mean \(\sum1/n^2=1\).
(l) Suppose a natural extension is not decreasing until \(x=10\), but is positive, continuous, and decreasing afterward. Explain whether the Integral Test can still be used.

Check the solution

(a) With \(u=\ln x\), the integral becomes \(\int_{\ln2}^{\infty}u^{-3}du\), which converges. The series converges.
(b) \(\int_2^\infty dx/(x\ln x)=\lim\limits_{b\to\infty}[\ln(\ln x)]_2^b=\infty\), so the series diverges.
(c) \(f(x)=1/(x+1)^2\) is positive, continuous, and decreasing, and \(\int_1^\infty dx/(x+1)^2=1/2\). The series converges.
(d) \(f(x)=x/(x^2+1)\) is positive and eventually decreasing. Since \(\int_2^\infty x/(x^2+1)dx=\tfrac12\lim\limits_{b\to\infty}\ln(b^2+1)-\tfrac12\ln5=\infty\), the series diverges.
(e) The extension \(f(x)=e^{-\sqrt x}\) is positive, continuous, and decreasing. Let \(u=\sqrt x\), so \(dx=2u\,du\); \(\int_1^\infty e^{-\sqrt x}dx=2\int_1^\infty ue^{-u}du\), which is finite. The series converges.
(f) Let \(u=\ln(\ln x)\), so \(du=dx/[x\ln x]\). The integral becomes \(\int_{\ln(\ln3)}^\infty du/u\), which diverges. Therefore the series diverges.
(g) The extension is not decreasing and the terms satisfy \(2+\sin n\ge1\), so they do not approach zero. The series diverges by the nth term test.
(h) \(\int_{N+1}^\infty x^{-2}dx=1/(N+1)\le R_N\le\int_N^\infty x^{-2}dx=1/N\).
(i) Require \(1/N<0.01\), so \(N>100\). The least guaranteed integer is \(101\).
(j) \(S_5=1+1/4+1/9+1/16+1/25\approx1.46361\). Hence \(S_5+1/6\le S\le S_5+1/5\), or approximately \(1.63028\le S\le1.66361\).
(k) The integral and series share convergence behavior but represent different accumulations. The series sum is \(\pi^2/6\), not \(1\).
(l) Yes. Apply the Integral Test to the tail beginning at \(n=10\); the finitely many earlier terms do not affect convergence.