How To Do The Integral Test

The Integral Test is a powerful tool in calculus used to determine the convergence or divergence of an infinite series by comparing it to the convergence or divergence of an improper integral. It's especially useful when dealing with series whose terms can be expressed as a function of a continuous variable. Understanding how to apply this test effectively requires a firm grasp of its conditions, the integration process, and interpreting the results.

Understanding the Integral Test

The Integral Test provides a way to connect the discrete world of series with the continuous world of integrals. The core idea is this: if you have a series whose terms are positive and decreasing, and you can find a continuous, decreasing function that matches those terms at integer values, then the convergence or divergence of the series is linked to the convergence or divergence of the integral of that function.

The Formal Statement:

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

• If ∫₁^∞ f(x) dx converges, then the series ∑_n=1^∞ a_n converges.
• If ∫₁^∞ f(x) dx diverges, then the series ∑_n=1^∞ a_n diverges.

Why Does This Work?

The intuition behind the Integral Test lies in visualizing the area under the curve of f(x). You can think of the terms a_n of the series as the heights of rectangles with width 1.

• Convergence: If the integral converges, it means the area under the curve f(x) is finite. The sum of the areas of the rectangles (representing the series) will also be finite since the rectangles are "trapped" under the curve (because f(x) is decreasing). Therefore, the series converges.

• Divergence: If the integral diverges, the area under the curve f(x) is infinite. The sum of the areas of the rectangles will also be infinite, as the rectangles extend at least as far as the curve. Therefore, the series diverges.

Prerequisites for Using the Integral Test

Before applying the Integral Test, it's absolutely crucial to verify that the function f(x) and the corresponding series term a_n satisfy the following three conditions:

1. Continuity: f(x) must be continuous on the interval [1, ∞). This means there are no breaks, jumps, or vertical asymptotes in the function's graph on this interval. If the function isn't continuous, the integral is not properly defined.

2. Positivity: f(x) must be positive on the interval [1, ∞). This ensures that the terms of the series are positive. If the terms are negative or alternating, the Integral Test cannot be directly applied. You might need to consider other tests, like the Alternating Series Test, in such cases.

3. Decreasing: f(x) must be decreasing on the interval [1, ∞). This means that the function's value must consistently decrease as x increases. This condition is essential for the comparison between the integral and the series to be valid. To prove that f(x) is decreasing, you can show that its derivative, f'(x), is negative on the interval [1, ∞).

Why are these conditions so important?

• Continuity: The integral ∫₁^∞ f(x) dx is defined based on the concept of area under a continuous curve. If f(x) has discontinuities (e.g., jumps or vertical asymptotes), the integral may not be properly defined or may not accurately represent the sum of the series.

• Positivity: The Integral Test relies on comparing the area under the curve f(x) to the sum of rectangular areas represented by the series terms. If f(x) takes on negative values, the area under the curve might cancel out, leading to an incorrect conclusion about the series' convergence or divergence.

• Decreasing: The decreasing condition ensures that the rectangles formed by the series terms either lie entirely below the curve (if the integral converges) or extend beyond the curve (if the integral diverges). This relationship is critical for the comparison between the integral and the series to hold. If f(x) is not decreasing, the area under the curve may not accurately reflect the sum of the series terms.

What if these conditions are not met?

If one or more of these conditions are not satisfied, the Integral Test cannot be used to determine the convergence or divergence of the series. In such cases, you'll need to consider other convergence tests, such as:

• The Comparison Test: Compare the series to another series whose convergence is known.
• The Limit Comparison Test: Compare the limit of the ratio of the series terms to another series whose convergence is known.
• The Ratio Test: Use the ratio of consecutive terms to determine convergence.
• The Root Test: Use the nth root of the absolute value of the terms to determine convergence.
• The Alternating Series Test: Specifically for alternating series (series with alternating signs).

It's vital to carefully analyze the series and the corresponding function f(x) to determine which test is most appropriate.

Steps to Perform the Integral Test

Here's a step-by-step guide to applying the Integral Test:

1. Identify the Series: Clearly state the infinite series ∑_n=1^∞ a_n you want to analyze.

2. Find the Corresponding Function: Determine the continuous function f(x) such that f(n) = a_n for all integers n ≥ 1. This often involves replacing n with x in the expression for a_n.

3. Verify the Conditions: Check that f(x) is continuous, positive, and decreasing on the interval [1, ∞). This is the most crucial step!

   • Continuity: Ensure that f(x) has no breaks, jumps, or vertical asymptotes on [1, ∞).
   • Positivity: Confirm that f(x) > 0 for all x in [1, ∞).
   • Decreasing: Show that f'(x) < 0 for all x in [1, ∞). This usually involves finding the derivative of f(x) and analyzing its sign.

4. Evaluate the Improper Integral: Calculate the improper integral ∫₁^∞ f(x) dx. Remember that an improper integral is defined as a limit:

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

   • Find the Antiderivative: Determine the antiderivative F(x) of f(x).
   • Evaluate the Limit: Calculate lim_t→∞ [F(t) - F(1)].

5. Interpret the Result:

   • If the integral converges (i.e., the limit exists and is a finite number): Then the series ∑_n=1^∞ a_n also converges.
   • If the integral diverges (i.e., the limit is infinite or does not exist): Then the series ∑_n=1^∞ a_n also diverges.

Important Notes:

• The Integral Test only tells you whether the series converges or diverges. It does not tell you the value to which the series converges.

• The lower limit of integration (1 in this case) can be changed to any value N ≥ 1, as long as f(x) satisfies the conditions of the test on the interval [N, ∞). Sometimes, starting the interval at a higher value makes it easier to satisfy the decreasing condition.

Examples of Applying the Integral Test

Let's illustrate the Integral Test with some examples:

Example 1: The p-series (p > 1)

Consider the series ∑_n=1^∞ 1/n^p, where p > 1. This is a p-series.

1. Identify the Series: ∑_n=1^∞ 1/n^p

2. Find the Corresponding Function: f(x) = 1/x^p = x^-p

3. Verify the Conditions:

   • Continuity: f(x) is continuous on [1, ∞) since x^p is continuous and non-zero for x ≥ 1.
   • Positivity: f(x) > 0 for all x in [1, ∞) since x^p is positive.
   • Decreasing: f'(x) = -p x^-p-1 = -p/ x^p+1. Since p > 1 and x ≥ 1, f'(x) < 0, so f(x) is decreasing.

4. Evaluate the Improper Integral:

   ∫₁^∞ x^-p dx = lim_t→∞ ∫₁^t x^-p dx

   The antiderivative of x^-p is x^-p+1 / (-p+1) = x^1-p / (1-p). Therefore,

   lim_t→∞ [t^1-p / (1-p) - 1^1-p / (1-p)] = lim_t→∞ [t^1-p / (1-p) - 1 / (1-p)]

   Since p > 1, 1-p < 0. Therefore, lim_t→∞ t^1-p = 0.

   So, the integral converges to 0 - 1/(1-p) = 1/(p-1).

5. Interpret the Result:

   Since the integral converges, the series ∑_n=1^∞ 1/n^p also converges for p > 1.

Example 2: The Harmonic Series (p = 1)

Consider the series ∑_n=1^∞ 1/n. This is the harmonic series.

1. Identify the Series: ∑_n=1^∞ 1/n

2. Find the Corresponding Function: f(x) = 1/x

3. Verify the Conditions:

   • Continuity: f(x) is continuous on [1, ∞).
   • Positivity: f(x) > 0 for all x in [1, ∞).
   • Decreasing: f'(x) = -1/x². Since x ≥ 1, f'(x) < 0, so f(x) is decreasing.

4. Evaluate the Improper Integral:

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

   The antiderivative of 1/x is ln(x). Therefore,

   lim_t→∞ [ln(t) - ln(1)] = lim_t→∞ ln(t) = ∞

5. Interpret the Result:

   Since the integral diverges, the series ∑_n=1^∞ 1/n also diverges.

Example 3: ∑_n=2^∞ 1/(n ln(n))

Notice the series starts at n=2. This is important!

1. Identify the Series: ∑_n=2^∞ 1/(n ln(n))

2. Find the Corresponding Function: f(x) = 1/(x ln(x))

3. Verify the Conditions:

   • Continuity: f(x) is continuous on [2, ∞) since x and ln(x) are continuous and non-zero on this interval (ln(x) is only zero at x=1).

   • Positivity: f(x) > 0 for all x in [2, ∞) since x and ln(x) are positive on this interval.

   • Decreasing: To show f(x) is decreasing, we can find f'(x) using the quotient rule:

     f'(x) = [ (0)(x ln(x)) - (1)(ln(x) + 1) ] / (x ln(x))² = -(ln(x) + 1) / (x ln(x))²

     Since ln(x) + 1 > 0 and (x ln(x))² > 0 for x ≥ 2, f'(x) < 0. Therefore, f(x) is decreasing.

4. Evaluate the Improper Integral:

   ∫₂^∞ 1/(x ln(x)) dx = lim_t→∞ ∫₂^t 1/(x ln(x)) dx

   To find the antiderivative, use the substitution u = ln(x), du = (1/x) dx:

   ∫ 1/(x ln(x)) dx = ∫ (1/u) du = ln|u| + C = ln|ln(x)| + C

   Therefore,

   lim_t→∞ [ln|ln(t)| - ln|ln(2)|] = lim_t→∞ ln(ln(t)) - ln(ln(2))

   As t approaches infinity, ln(t) also approaches infinity, and ln(ln(t)) approaches infinity.

5. Interpret the Result:

   Since the integral diverges, the series ∑_n=2^∞ 1/(n ln(n)) also diverges.

Common Mistakes to Avoid

• Forgetting to Verify the Conditions: This is the most common mistake. Always check continuity, positivity, and decreasing behavior before applying the Integral Test.
• Incorrectly Evaluating the Integral: Double-check your integration and limit calculations.
• Applying the Test When It's Not Appropriate: The Integral Test is not a universal test. Choose the appropriate convergence test based on the characteristics of the series.
• Confusing Convergence/Divergence of the Integral with the Value of the Series: The integral test only tells you IF a series converges. If it converges, the VALUE of the integral is NOT necessarily the value the series converges to.
• Ignoring the Lower Limit of Integration: Ensure the conditions for the integral test are met on the entire interval of integration. Adjust the lower limit if necessary (and appropriate).

Conclusion

The Integral Test is a valuable technique for determining the convergence or divergence of infinite series, particularly when the series terms can be represented by a continuous, positive, and decreasing function. By carefully verifying the conditions of the test and accurately evaluating the corresponding improper integral, you can effectively apply this tool to analyze the behavior of a wide range of series. Remember to choose the most appropriate convergence test based on the characteristics of the series you are analyzing. Mastery of the Integral Test will significantly enhance your understanding of infinite series and their applications in calculus and beyond.