Integral Test For Convergence Examples

Integral Test for Convergence: Examples and Deep Dive

The integral test is a powerful tool in calculus for determining the convergence or divergence of an infinite series. It provides a direct link between the sum of a series and the area under a curve, offering an elegant and often simpler approach than other convergence tests. This article will delve into the intricacies of the integral test, providing a comprehensive understanding through detailed explanations, worked examples, and a frequently asked questions section. Understanding the integral test is crucial for anyone studying calculus, especially for those tackling challenging problems involving infinite series.

Understanding the Integral Test: A Foundation

The integral test establishes a connection between the convergence of a series and the convergence of an improper integral. Specifically, consider a series ∑_n=1^∞ f(n), where f(x) is a continuous, positive, and decreasing function for x ≥ 1. The integral test states:

• If ∫₁^∞ f(x) dx converges, then ∑_n=1^∞ f(n) also converges.
• If ∫₁^∞ f(x) dx diverges, then ∑_n=1^∞ f(n) also diverges.

Crucially, the function f(x) must satisfy three conditions: it must be continuous, positive, and decreasing for x ≥ 1. If any of these conditions are not met, the integral test is not applicable. Let's examine each condition in detail:

• Continuity: The function must be continuous within the interval of integration. Discontinuities can disrupt the relationship between the series and the integral.

• Positivity: The function must be positive for all x ≥ 1. This ensures that both the series terms and the integral represent areas above the x-axis. A negative function would lead to incorrect conclusions.

• Decreasing: The function must be monotonically decreasing (or at least eventually decreasing) for x ≥ 1. This ensures that the rectangles used to approximate the area under the curve are consistently larger than the corresponding terms in the series (or vice versa).

Applying the Integral Test: Step-by-Step Guide

Let's break down the application of the integral test into manageable steps:

1. Identify the function f(x): Extract the function f(x) from the given series ∑_n=1^∞ f(n). Simply replace 'n' with 'x'.

2. Verify the conditions: Check if f(x) is continuous, positive, and decreasing for x ≥ 1. If not, the integral test cannot be used.

3. Evaluate the improper integral: Calculate the improper integral ∫₁^∞ f(x) dx. This often involves using limits and techniques of integration.

4. Determine convergence or divergence: Based on the result of the integral:

   • If the integral converges (i.e., evaluates to a finite value), the series also converges.
   • If the integral diverges (i.e., evaluates to ∞), the series also diverges.

5. State the conclusion: Clearly state whether the series converges or diverges based on your findings.

Examples of the Integral Test in Action

Let's illustrate the integral test with several examples, demonstrating its application and highlighting nuances:

Example 1: The Harmonic Series

Consider the harmonic series ∑_n=1^∞ (1/n). Let f(x) = 1/x. This function is continuous, positive, and decreasing for x ≥ 1.

The improper integral is:

∫₁^∞ (1/x) dx = lim_b→∞ [ln|x|]₁^b = lim_b→∞ (ln|b| - ln|1|) = ∞

Since the integral diverges, the harmonic series also diverges.

Example 2: p-series

The p-series is given by ∑_n=1^∞ (1/n^p), where p is a constant. Let's use the integral test to determine its convergence.

Let f(x) = 1/x^p. This function is continuous, positive, and decreasing for x ≥ 1, provided p > 0.

The improper integral is:

∫₁^∞ (1/x^p) dx = lim_b→∞ [(x^1-p)/(1-p)]₁^b

This integral converges if 1 - p < 0, which means p > 1. If p ≤ 1, the integral diverges.

Therefore, the p-series converges if p > 1 and diverges if p ≤ 1. This is a crucial result in the study of infinite series.

Example 3: A More Complex Example

Consider the series ∑_n=2^∞ (1/(nln(n))). Let f(x) = 1/(xln(x)). This function is continuous, positive, and decreasing for x ≥ 2.

The improper integral is:

∫₂^∞ (1/(x*ln(x))) dx = lim_b→∞ [ln|ln|x||]₂^b = ∞

Since the integral diverges, the series ∑_n=2^∞ (1/(n*ln(n))) also diverges. Note that the lower limit of integration changes to reflect the starting point of the series.

Example 4: A Series That Fails the Conditions

Consider the series ∑_n=1^∞ (-1)ⁿ/n. This is an alternating series, and the integral test does not apply because the terms are not all positive. We need a different test, like the Alternating Series Test, to assess convergence.

When the Integral Test is Inapplicable

It's crucial to understand that the integral test is not a universal solution for determining series convergence. Here are some situations where it's inappropriate:

• Non-positive functions: The function f(x) must be positive for x ≥ 1. If the series contains negative terms, the integral test cannot be applied.

• Non-decreasing functions: The function f(x) must be decreasing (or eventually decreasing) for x ≥ 1. If the function is increasing or oscillates, the integral test is invalid.

• Discontinuous functions: The integral test requires a continuous function. Discontinuities can invalidate the comparison between the series and the integral.

In cases where the integral test is not applicable, other convergence tests, such as the comparison test, limit comparison test, ratio test, root test, or alternating series test, should be considered.

Frequently Asked Questions (FAQ)

Q1: Can the integral test be used for series that start at an index other than 1?

A1: Yes, the integral test can be adapted. The integral's lower limit will simply change to match the starting index of the series. For example, if the series starts at n=2, the integral would be from 2 to ∞.

Q2: What if the integral is difficult or impossible to evaluate?

A2: If the integral is too complex to evaluate directly, you might be able to use limit comparison tests or other approaches to determine its convergence or divergence. Sometimes, comparing the integral to a known convergent or divergent integral can provide an answer.

Q3: Is the integral test always the best method for determining convergence?

A3: No. The integral test is a valuable tool, but its applicability depends on the properties of the series and the ease of evaluating the corresponding integral. Other convergence tests might be more efficient or appropriate in certain situations. Choose the most suitable test for each individual problem.

Q4: Can the integral test determine the sum of a convergent series?

A4: No. The integral test only determines whether a series converges or diverges. It does not provide information about the sum of the series if it converges. Finding the exact sum often requires different techniques.

Conclusion

The integral test provides a powerful and intuitive method for determining the convergence or divergence of infinite series. By linking the series to the area under a curve, it offers an elegant approach that's often easier to apply than other tests. However, it is crucial to remember the conditions that must be met and to consider alternative methods when the integral test is inapplicable. Understanding the integral test thoroughly enhances your problem-solving skills in calculus and deepens your understanding of infinite series. Mastering this test is a significant step toward achieving fluency in advanced calculus concepts. Remember to always carefully verify the conditions before applying the integral test; understanding these conditions is as important as the test itself.