Integral Test For Convergence Practice Problems

Integral Test for Convergence: Practice Problems and Solutions

The integral test is a powerful tool in your calculus arsenal, offering a straightforward method to determine the convergence or divergence of infinite series. This method links the behavior of a series to the convergence or divergence of an improper integral. Mastering the integral test requires both a solid understanding of the underlying theory and plenty of practice. This article dives deep into the integral test, providing a comprehensive explanation alongside a range of practice problems with detailed solutions.

Understanding the Integral Test

Before we tackle problems, let's review the core principles of the integral test. The test states that if f(x) is a continuous, positive, and decreasing function on the interval [1, ∞) such that a_n = f(n) for all n ≥ 1, then the infinite series Σ a_n converges if and only if the improper integral ∫₁^∞ f(x) dx converges.

Key Conditions for Applying the Integral Test:

• Continuous: The function f(x) must be continuous on the interval [1, ∞). This means there are no breaks or jumps in the graph.
• Positive: The function f(x) must be positive on the interval [1, ∞). This means the function's values are always greater than zero.
• Decreasing: The function f(x) must be decreasing on the interval [1, ∞). This means that as x increases, f(x) decreases.

When the Integral Test Fails:

The integral test doesn't always apply. If the function f(x) fails to meet any of the three conditions (continuous, positive, decreasing), then the integral test cannot be used to determine the convergence or divergence of the series. In such cases, other convergence tests, such as the comparison test, limit comparison test, or ratio test, might be necessary.

Practice Problems: From Simple to Advanced

Let's work through a series of practice problems, starting with simpler examples and gradually increasing the complexity. Each problem will detail the steps needed to apply the integral test and interpret the results.

Problem 1: A Basic Example

Determine the convergence or divergence of the series Σ (1/n²) from n=1 to ∞.

Solution:

1. Define f(x): Let f(x) = 1/x².

2. Check Conditions:

   • Continuous: f(x) is continuous on [1, ∞).
   • Positive: f(x) is positive on [1, ∞).
   • Decreasing: f(x) is decreasing on [1, ∞). The derivative, f'(x) = -2/x³, is negative for x > 0.

3. Evaluate the Integral:

   ∫₁^∞ (1/x²) dx = lim_b→∞ ∫₁^b (1/x²) dx = lim_b→∞ [-1/x]₁^b = lim_b→∞ (-1/b + 1) = 1

4. Conclusion: Since the integral converges (to 1), the series Σ (1/n²) also converges. This is a well-known p-series (p=2 > 1), which always converges.

Problem 2: A Slightly More Challenging Case

Determine the convergence or divergence of the series Σ (1/(n*ln(n))) from n=2 to ∞ (Note: the series starts at n=2 because ln(1) = 0).

Solution:

1. Define f(x): Let f(x) = 1/(xln(x))*.

2. Check Conditions:

   • Continuous: f(x) is continuous on [2, ∞).
   • Positive: f(x) is positive on [2, ∞).
   • Decreasing: f'(x) = -(ln(x) + 1)/(x²(ln(x))²) < 0* for x > 2. Thus, f(x) is decreasing on [2, ∞).

3. Evaluate the Integral:

   ∫₂^∞ (1/(xln(x))) dx = lim_b→∞ ∫₂^b (1/(xln(x))) dx

   Use u-substitution: let u = ln(x), so du = (1/x)dx. The limits of integration become ln(2) and ln(b).

   = lim_b→∞ ∫_ln(2)^ln(b) (1/u) du = lim_b→∞ [ln|u|]_ln(2)^ln(b) = lim_b→∞ (ln(ln(b)) - ln(ln(2))) = ∞

4. Conclusion: Since the integral diverges, the series Σ (1/(n*ln(n))) also diverges.

Problem 3: Dealing with a More Complex Function

Determine the convergence or divergence of the series Σ (e^-n/n) from n=1 to ∞.

Solution:

1. Define f(x): Let f(x) = e^-x/x.

2. Check Conditions:

   • Continuous: f(x) is continuous on [1, ∞).
   • Positive: f(x) is positive on [1, ∞).
   • Decreasing: f'(x) = -(xe^-x + e^-x)/x² < 0 for x ≥ 1. Thus f(x) is decreasing on [1, ∞).

3. Evaluate the Integral: This integral requires integration by parts. Let's outline the steps:

   ∫₁^∞ (e^-x/x) dx This integral is difficult to solve analytically. However, we can use the fact that e^-x/x < e^-x for x ≥ 1. Since ∫₁^∞ e^-x dx converges (to e^-1), by the comparison test, ∫₁^∞ (e^-x/x) dx also converges.

4. Conclusion: Although we didn't directly solve the integral, the comparison test, combined with the integral test's requirement of a decreasing positive function, lets us conclude that the series Σ (e^-n/n) converges.

Problem 4: A Case Where the Integral Test Doesn't Apply Directly

Consider the series Σ (-1)ⁿ/n from n=1 to ∞. This is an alternating series.

Solution:

The integral test does not apply directly because the terms of the series are not all positive. The integral test only works for series with positive terms. To analyze this series, you'd use the Alternating Series Test which considers the alternating nature of the terms.

Problem 5: Understanding the Importance of Checking Conditions

Consider the series Σ (1/n) from n=1 to ∞.

Solution:

Let f(x) = 1/x. While this function is positive and continuous on the interval [1, ∞), it does not satisfy the decreasing condition if we start the series at n=1 (though technically, if you have a finite number of non-decreasing terms and the tail of the series is monotonically decreasing, that does not violate the integral test). However, let's show the process. The integral is:

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

This means the integral diverges, indicating that the harmonic series Σ (1/n) also diverges.

Advanced Techniques and Considerations

The integral test, while powerful, requires careful consideration of the function's properties. Here are some additional points to keep in mind:

• Using Comparison Tests in Conjunction with the Integral Test: As seen in Problem 3, sometimes direct evaluation of the integral is difficult or impossible. In such cases, comparison tests can be invaluable in determining convergence or divergence.

• Handling Discontinuous Functions: If the function has a finite number of discontinuities, you can often split the integral into multiple parts, handling the discontinuities carefully.

• Numerical Methods for Approximating Integrals: For complex integrals that cannot be solved analytically, numerical methods (such as the trapezoidal rule or Simpson's rule) can be used to approximate the value of the integral, providing an indication of convergence or divergence.

Conclusion

The integral test is an important tool for determining the convergence or divergence of infinite series. By understanding its conditions and applying it strategically, coupled with other convergence tests when appropriate, you can confidently analyze a wide range of series. Remember to always carefully check the three conditions (continuity, positivity, and decreasing nature) before applying the test. Consistent practice is key to mastering this technique. The problems above represent a range of difficulty, from basic examples to more challenging scenarios that require additional techniques. Continue practicing to hone your problem-solving skills and deepen your understanding of infinite series and the integral test.