How To Use The Integral Test

Let's explore the integral test, a powerful tool in calculus for determining the convergence or divergence of infinite series. This test provides a way to relate the behavior of a series to the behavior of an integral, often simplifying the analysis considerably. By understanding the integral test, you can unlock a new level of understanding when dealing with infinite sums.

Understanding the Integral Test

The integral test bridges the gap between infinite series and improper integrals. At its core, it states that if we have a series whose terms are defined by a positive, continuous, and decreasing function, then the convergence or divergence of the series is directly linked to the convergence or divergence of the integral of that function over the same interval.

Formal Statement:

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1, ∞). Consider the series Σ a_n, where a_n = f(n) for all integers n ≥ 1. Then:

• If the improper integral ∫₁^∞ f(x) dx converges, the series Σ a_n also converges.
• If the improper integral ∫₁^∞ f(x) dx diverges, the series Σ a_n also diverges.

Key Requirements for the Integral Test:

• Continuity: f(x) must be continuous on the interval [1, ∞). This means there are no breaks, jumps, or undefined points.
• Positivity: f(x) must be positive on the interval [1, ∞). All values of the function must be greater than zero.
• Decreasing: f(x) must be decreasing on the interval [1, ∞). This means that as x increases, the value of f(x) either decreases or remains constant (but it is better if it strictly decreases).

Why Does the Integral Test Work?

The integral test's validity stems from the geometric relationship between the area under the curve of f(x) and the sum of the terms in the series. Imagine constructing rectangles whose widths are 1 and whose heights are given by the function values f(1), f(2), f(3), and so on.

• Convergence: If the integral converges, it means the area under the curve f(x) from 1 to infinity is finite. We can visually see that the sum of the areas of the rectangles (representing the series) is less than the area under the curve plus the first term, f(1). Since the integral is finite, the sum of the areas of the rectangles must also be finite, implying that the series converges.
• Divergence: If the integral diverges, the area under the curve is infinite. We can see that the sum of the areas of the rectangles is greater than the area under the curve. Since the integral is infinite, the sum of the areas of the rectangles must also be infinite, indicating that the series diverges.

Steps to Apply the Integral Test

Here's a breakdown of the steps involved in applying the integral test:

1. Verify the Conditions: The most crucial step is to verify that the function f(x), corresponding to the series term a_n, satisfies the three conditions: continuity, positivity, and decreasing nature on the interval [1, ∞). If any of these conditions are not met, the integral test cannot be used. You'll need to explore other convergence/divergence tests.

2. Define the Function: Replace n in the general term a_n of the series with x to obtain the function f(x). For example, if a_n = 1/n², then f(x) = 1/x².

3. Evaluate the Improper Integral: Calculate the improper integral ∫₁^∞ f(x) dx. Remember that an improper integral is an integral where one or both limits of integration are infinite, or the integrand has a discontinuity within the interval of integration. This typically involves evaluating a limit:

   ∫₁^∞ f(x) dx = lim_t→∞ ∫₁^t f(x) dx

   Evaluate the definite integral ∫₁^t f(x) dx, and then find the limit as t approaches infinity.

4. Determine Convergence or Divergence: Based on the result of the improper integral:

   • If the limit exists and is a finite number, the integral converges, and therefore the series also converges.
   • If the limit does not exist (e.g., it approaches infinity) or is infinite, the integral diverges, and therefore the series also diverges.

5. State Your Conclusion: Clearly state whether the series converges or diverges based on the integral test.

Examples of Using the Integral Test

Let's illustrate the integral test with a few examples.

Example 1: The Series Σ 1/n² (p-series with p = 2)

1. Define the Function: f(x) = 1/x²

2. Verify the Conditions:

   • Continuity: f(x) = 1/x² is continuous for all x ≥ 1.
   • Positivity: f(x) = 1/x² is positive for all x ≥ 1.
   • Decreasing: The derivative of f(x) is f'(x) = -2/x³, which is negative for all x ≥ 1. Therefore, f(x) is decreasing.

3. Evaluate the Improper Integral:

   ∫₁^∞ (1/x²) dx = lim_t→∞ ∫₁^t (1/x²) dx = lim_t→∞ [-1/x]₁^t = lim_t→∞ (-1/t - (-1/1)) = lim_t→∞ (1 - 1/t) = 1

4. Determine Convergence or Divergence: The integral converges to 1.

5. Conclusion: By the integral test, the series Σ 1/n² converges.

Example 2: The Series Σ 1/n

This is the famous harmonic series.

1. Define the Function: f(x) = 1/x

2. Verify the Conditions:

   • Continuity: f(x) = 1/x is continuous for all x ≥ 1.
   • Positivity: f(x) = 1/x is positive for all x ≥ 1.
   • Decreasing: The derivative of f(x) is f'(x) = -1/x², which is negative for all x ≥ 1. Therefore, f(x) is decreasing.

3. Evaluate the Improper Integral:

   ∫₁^∞ (1/x) dx = lim_t→∞ ∫₁^t (1/x) dx = lim_t→∞ [ln(x)]₁^t = lim_t→∞ (ln(t) - ln(1)) = lim_t→∞ ln(t) = ∞

4. Determine Convergence or Divergence: The integral diverges to infinity.

5. Conclusion: By the integral test, the series Σ 1/n diverges.

Example 3: The Series Σ n/eⁿ

1. Define the Function: f(x) = x/e^x

2. Verify the Conditions:

   • Continuity: f(x) = x/e^x is continuous for all x ≥ 1.
   • Positivity: f(x) = x/e^x is positive for all x ≥ 1.
   • Decreasing: We need to find the derivative and check its sign. f'(x) = (e^x - xe^x) / (e^x)² = (1-x) / e^x For x > 1, f'(x) is negative. Thus, f(x) is decreasing for x ≥ 1.

3. Evaluate the Improper Integral:

   ∫₁^∞ (x/e^x) dx = lim_t→∞ ∫₁^t (x/e^x) dx

   We need to use integration by parts: Let u = x, dv = e^-x dx. Then du = dx, v = -e^-x.

   ∫ x/e^x dx = -x e^-x - ∫ -e^-x dx = -x e^-x - e^-x = -(x + 1)/e^x

   So, lim_t→∞ ∫₁^t (x/e^x) dx = lim_t→∞ [-(x + 1)/e^x]₁^t = lim_t→∞ [-(t + 1)/e^t - (- (1 + 1)/e¹)] = lim_t→∞ [-(t + 1)/e^t + 2/e]

   Using L'Hopital's rule on the first term: lim_t→∞ -(t + 1)/e^t = lim_t→∞ -1/e^t = 0

   Therefore, the integral equals 0 + 2/e = 2/e

4. Determine Convergence or Divergence: The integral converges to 2/e.

5. Conclusion: By the integral test, the series Σ n/eⁿ converges.

When NOT to Use the Integral Test

The integral test is a powerful tool, but it's not always the best choice. Here are some situations where you should consider using other convergence/divergence tests:

• The conditions are not met: If f(x) is not continuous, positive, or decreasing on the interval [1, ∞), the integral test is not applicable. Using it in these cases will lead to incorrect conclusions.
• The integral is difficult to evaluate: Sometimes, the integral ∫₁^∞ f(x) dx is very difficult or impossible to evaluate. In such cases, other tests like the comparison test, limit comparison test, ratio test, or root test might be more suitable.
• Series with alternating signs: The integral test is designed for series with positive terms. For series with alternating signs, the alternating series test is more appropriate.
• Geometric series: For geometric series, it's much easier to directly apply the geometric series test.
• p-series: While the integral test can be used for p-series, it's often faster to simply remember the p-series test, which states that Σ 1/n^p converges if p > 1 and diverges if p ≤ 1.

Common Mistakes to Avoid

• Forgetting to verify the conditions: This is the most common mistake. Always check that f(x) is continuous, positive, and decreasing before applying the integral test.
• Incorrectly evaluating the integral: Make sure to evaluate the integral correctly, paying attention to improper integral techniques and any necessary integration by parts or u-substitution.
• Confusing convergence/divergence of the integral with the value of the series: The integral test tells you whether the series converges or diverges, but it does not tell you the value to which the series converges (except in rare cases where you can find the exact sum). The value of the integral and the value of the series are generally different.
• Applying the test to a finite sum: The integral test is for infinite series. It doesn't make sense to apply it to a finite sum of terms.

Other Convergence Tests: A Brief Overview

Knowing other convergence tests expands your ability to analyze series. Here are a few common ones:

• The Comparison Test: If 0 ≤ a_n ≤ b_n for all n, then:
  • If Σ b_n converges, then Σ a_n converges.
  • If Σ a_n diverges, then Σ b_n diverges.
• The Limit Comparison Test: If lim_n→∞ (a_n/b_n) = c, where 0 < c < ∞, then Σ a_n and Σ b_n either both converge or both diverge.
• The Ratio Test: Let L = lim_n→∞ |a_n+1/a_n|. Then:
  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.
• The Root Test: Let L = lim_n→∞ (a_n)^1/n. Then:
  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.
• The Alternating Series Test: For an alternating series Σ (-1)ⁿ b_n (where b_n > 0), if b_n is decreasing and lim_n→∞ b_n = 0, then the series converges.
• The Geometric Series Test: The geometric series Σ arⁿ converges if |r| < 1 and diverges if |r| ≥ 1.

Conclusion

The integral test is a powerful technique for determining the convergence or divergence of infinite series. By relating the behavior of a series to the behavior of an integral, it provides a valuable tool for analyzing infinite sums. Remember to carefully verify the conditions before applying the test and be mindful of common mistakes. Mastering the integral test, along with other convergence tests, will significantly enhance your understanding of infinite series and their applications. Practice applying the integral test to various examples to solidify your understanding and develop your problem-solving skills in calculus. Remember that choosing the right convergence test is an art form that comes with practice and a deep understanding of the properties of different series. Good luck!