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Integral Test
for Convergence

Use improper integrals to determine whether an infinite series converges or diverges: bridging the continuous and the discrete.

AP Calculus BC

Why Do Infinite Series Matter?

An infinite series \(\displaystyle\sum_{n=1}^{\infty} a_n\) adds up infinitely many terms. The central question: does that sum settle on a finite number (converge) or grow without bound (diverge)?

Converges
The partial sums \(S_N = \sum_{n=1}^{N} a_n\) approach a finite limit as \(N \to \infty\).
Diverges
The partial sums grow without bound or oscillate: no finite limit exists.

The Big Idea: Rectangles Under a Curve

Imagine plotting each term \(a_n\) as the height of a rectangle of width 1. Those rectangles sit under or over a smooth curve \(f(x)\) where \(f(n) = a_n\).

x y 1 2 3 4 5 6 7 8 = \(a_n\) (rectangle heights) = \(f(x)\) (continuous curve)

Three Prerequisites You Must Check

The Integral Test only applies when the function \(f\) satisfies all three conditions on \([1, \infty)\):

Continuous
\(f(x)\) has no jumps, holes, or vertical asymptotes on the interval.
Positive
\(f(x) > 0\) for all \(x \geq 1\). Every term of the series is positive.
Decreasing
\(f'(x) < 0\) for all \(x \geq 1\). The terms shrink as \(n\) increases.

The Integral Test: Formal Statement

The test does not tell you the value of the series. It only tells you whether the series converges. The integral and the series are companions: they share the same fate.

Series converges or diverges same fate Improper Integral converges or diverges

Why Does It Work?

Because \(f\) is decreasing, the rectangles give us upper and lower bounds for the integral:

1
Left rectangles overshoot: each rectangle's height \(a_n = f(n)\) exceeds the curve on \([n, n+1]\) because \(f\) is decreasing.
2
Right rectangles undershoot: each rectangle's height \(a_{n+1}\) is below the curve on \([n, n+1]\).
3
Squeeze: if the integral converges, the series is squeezed to a finite value. If the integral diverges, the series is pushed to infinity.

Convergence vs Value

A critical distinction students often miss:

Integral Test Tells You Integral Test Does NOT Tell You
Convergence ✓ Whether the series converges or diverges
Exact Sum ✕ The actual value \(\sum a_n\) equals
Equality ✕ \(\sum a_n = \int_1^{\infty} f(x)\,dx\)

The \(p\)-Series Result

The Integral Test gives us the famous \(p\)-series result as an immediate consequence:

\(p = 2\): Converges
\(\int_1^{\infty} x^{-2}\,dx = 1 < \infty\)
\(p = 1\): Diverges
\(\int_1^{\infty} x^{-1}\,dx = \ln x \Big|_1^{\infty} = \infty\)

How to Apply the Integral Test

1
Identify \(f(x)\): write the continuous version of \(a_n\) by replacing \(n\) with \(x\).
2
Check CPD: verify \(f\) is Continuous, Positive, and Decreasing on some \([N, \infty)\).
3
Evaluate the integral: compute \(\displaystyle\int_N^{\infty} f(x)\,dx = \lim_{t\to\infty} \int_N^{t} f(x)\,dx\).
4
Conclude: if the integral is finite, the series converges. If infinite, it diverges.

Example 1: \(\displaystyle\sum \frac{1}{n^2}\)

Determine whether \(\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2}\) converges or diverges.

1
Let \(f(x) = \dfrac{1}{x^2}\). Then \(f(n) = a_n = \dfrac{1}{n^2}\).
2
CPD check: On \([1,\infty)\), \(f\) is continuous, positive, and decreasing (since \(f'(x) = -2/x^3 < 0\)).
3
Evaluate: \(\displaystyle\int_1^{\infty} \frac{1}{x^2}\,dx = \lim_{t\to\infty}\left[-\frac{1}{x}\right]_1^t = \lim_{t\to\infty}\left(-\frac{1}{t}+1\right) = 1\).
4
Conclusion: The integral converges to 1, so the series converges by the Integral Test.

Example 2: \(\displaystyle\sum \frac{1}{\sqrt{n}}\)

Determine whether \(\displaystyle\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}\) converges or diverges.

1
Let \(f(x) = \dfrac{1}{\sqrt{x}} = x^{-1/2}\). CPD holds on \([1,\infty)\).
2
Evaluate: \(\displaystyle\int_1^{\infty} x^{-1/2}\,dx = \lim_{t\to\infty}\left[2\sqrt{x}\right]_1^t = \lim_{t\to\infty}(2\sqrt{t} - 2) = \infty\).
3
Conclusion: The integral diverges, so the series diverges.

Example 3: \(\displaystyle\sum \frac{1}{n^2+1}\)

Determine whether \(\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2+1}\) converges or diverges.

1
Let \(f(x) = \dfrac{1}{x^2+1}\). CPD: continuous (no asymptotes), positive, and decreasing on \([1,\infty)\).
2
Evaluate: \(\displaystyle\int_1^{\infty} \frac{1}{x^2+1}\,dx = \lim_{t\to\infty}\Big[\arctan(x)\Big]_1^t = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}\).
3
Conclusion: The integral converges to \(\pi/4\), so the series converges.

Visualizing \(\displaystyle\frac{1}{x^2+1}\)

x y 1.0 0.5 1 2 3 4 5 6 7 Area = \(\pi/4\) (integral) Rectangles = \(\sum a_n\)

Example 4: \(\displaystyle\sum \frac{\ln n}{n}\)

Determine whether \(\displaystyle\sum_{n=2}^{\infty} \frac{\ln n}{n}\) converges or diverges.

1
Let \(f(x) = \dfrac{\ln x}{x}\). CPD: continuous and positive on \([2,\infty)\). Decreasing for \(x \geq 3\) (verify: \(f'(x) = \frac{1-\ln x}{x^2} < 0\) when \(x > e\)).
2
Evaluate: Use \(u\)-sub: \(u = \ln x\), \(du = \frac{1}{x}\,dx\). \(\displaystyle\int_2^{\infty} \frac{\ln x}{x}\,dx = \lim_{t\to\infty} \frac{(\ln t)^2}{2} - \frac{(\ln 2)^2}{2} = \infty\).
3
Conclusion: The integral diverges, so \(\sum \frac{\ln n}{n}\) diverges.

Common Mistakes to Avoid

✕ Forgetting CPD
Applying the test to a function that is not decreasing or not positive on the interval. Always verify all three conditions.
✕ Equating Sum to Integral
Writing \(\sum a_n = \int f(x)\,dx\). The integral gives the convergence behavior, not the sum's value.
✕ Using Divergent Test Backwards
Thinking \(a_n \to 0\) means convergence. The harmonic series proves this wrong: \(\frac{1}{n} \to 0\) but \(\sum \frac{1}{n} = \infty\).

Series Convergence Tests at a Glance

Test When to Use Conclusion
Divergence Test \(a_n \not\to 0\) Diverges only
Integral Test \(f\) is CPD on \([1,\infty)\) Conv. or div.
\(p\)-Series \(\sum \frac{1}{n^p}\) \(p > 1\): conv.; \(p \leq 1\): div.
Comparison Can bound \(a_n\) by known series Conv. or div.
Ratio Test Factorials, exponentials \(L < 1\): conv.; \(L > 1\): div.

Question 1 of 5

Which of the following is not a required condition for using the Integral Test?

Question 2 of 5

Using the Integral Test, determine the behavior of \(\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3/2}}\).

Question 3 of 5

If \(\displaystyle\int_1^{\infty} f(x)\,dx = 5\), what can you conclude about \(\displaystyle\sum_{n=1}^{\infty} f(n)\)?

Question 4 of 5

For \(\displaystyle\sum_{n=1}^{\infty} \frac{1}{n \ln n}\), starting at \(n = 2\), what does the Integral Test reveal?

Question 5 of 5

Which series can not be tested using the Integral Test?