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AP Calculus BC 10.4 Integral Test for Convergence Study Notes - New Syllabus

AP Calculus BC 10.4 Integral Test for Convergence Study Notes- New syllabus

AP Calculus BC 10.4 Integral Test for Convergence Study Notes – AP Calculus BC-  per latest AP Calculus BC Syllabus.

LEARNING OBJECTIVE

  • Applying limits may allow us to determine the finite sum of infinitely many terms.

Key Concepts: 

  • Integral Test for Convergence

AP Calculus BC-Concise Summary Notes- All Topics

AP Calculus BC Exam Questions MCQs and FRQs

Integral Test for Convergence

Integral Test for Convergence

The Integral Test is a method used to determine whether an infinite series converges or diverges by comparing it to an improper integral.

Statement of the Test:

Let \( f(x) \) be a continuous, positive, and decreasing function for \( x \geq 1 \), and suppose \( a_n = f(n) \). Then:

  • If \( \displaystyle \int_{1}^{\infty} f(x) \, dx \) converges (has a finite value), then the series \( \displaystyle \sum_{n=1}^\infty a_n \) converges.
  • If \( \displaystyle \int_{1}^{\infty} f(x) \, dx \) diverges (is infinite), then the series \( \displaystyle \sum_{n=1}^\infty a_n \) diverges.

Conditions for Applying the Integral Test:

  1. Continuity: \( f(x) \) must be continuous for \( x \geq 1 \).
  2. Positivity: \( f(x) > 0 \) for \( x \geq 1 \).
  3. Decreasing Nature: \( f'(x) < 0 \) for \( x \geq 1 \).

Important Note:

  • The Integral Test works only when these conditions are met. If any condition fails, the test cannot be applied.
  • The test provides information about convergence/divergence, not the exact sum of the series.
  • It is often used for \( p \)-series and series with functions that can be integrated easily.
  • For borderline cases, other tests (comparison, limit comparison) may be more practical.

Example

Test the convergence of \( \displaystyle \sum_{n=1}^\infty \dfrac{1}{n^2} \) using the Integral Test.

▶️ Answer/Explanation

Here, \( a_n = \dfrac{1}{n^2} \) and \( f(x) = \dfrac{1}{x^2} \).

Check conditions: \( f(x) \) is continuous, positive, and decreasing for \( x \geq 1 \).

Evaluate \( \displaystyle \int_{1}^{\infty} \dfrac{1}{x^2} \, dx \):

\( \displaystyle \int_{1}^{\infty} x^{-2} \, dx = \left[ -x^{-1} \right]_{1}^{\infty} = 0 – (-1) = 1 \)

Since the improper integral converges to 1, the series also converges.

Final Answer: Convergent.

Example

Test the convergence of \( \displaystyle \sum_{n=1}^\infty \dfrac{1}{n} \) using the Integral Test.

▶️ Answer/Explanation

Here, \( a_n = \dfrac{1}{n} \) and \( f(x) = \dfrac{1}{x} \).

Check conditions: \( f(x) \) is continuous, positive, and decreasing for \( x \geq 1 \).

Evaluate \( \displaystyle \int_{1}^{\infty} \dfrac{1}{x} \, dx \):

\( \displaystyle \int_{1}^{\infty} x^{-1} \, dx = \left[ \ln x \right]_{1}^{\infty} = \infty \)

Since the improper integral diverges, the series also diverges.

Final Answer: Divergent.

Example

Test the convergence of \( \displaystyle \sum_{n=2}^\infty \dfrac{1}{n(\ln n)^2} \) using the Integral Test.

▶️ Answer/Explanation

Here, \( a_n = \dfrac{1}{n(\ln n)^2} \) and \( f(x) = \dfrac{1}{x(\ln x)^2} \).

Check conditions: \( f(x) \) is continuous, positive, and decreasing for \( x \geq 2 \).

Let \( u = \ln x \), \( du = \dfrac{1}{x} \, dx \).

\( \displaystyle \int_{2}^{\infty} \dfrac{1}{x(\ln x)^2} \, dx = \int_{\ln 2}^{\infty} u^{-2} \, du = \left[ -u^{-1} \right]_{\ln 2}^{\infty} = 0 – \left( -\dfrac{1}{\ln 2} \right) = \dfrac{1}{\ln 2} \)

Since the improper integral converges to \( \dfrac{1}{\ln 2} \), the series converges.

Final Answer: Convergent.

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