Understanding the Integral Test- A Guide to Determining Eligibility for Application
How do I know if I can apply the integral test? The integral test is a powerful tool in calculus that helps determine the convergence or divergence of an infinite series. It is particularly useful when dealing with series that involve continuous functions. In this article, we will explore the conditions under which the integral test can be applied and provide a step-by-step guide to help you determine its applicability to your specific series.
The integral test can be applied to a series of the form \(\sum_{n=1}^{\infty} a_n\), where \(a_n\) is a positive, continuous, and decreasing function of \(n\). To determine if the integral test is applicable, follow these steps:
1. Check for Positivity: Ensure that the function \(a_n\) is positive for all \(n\). If \(a_n\) is not positive, the integral test cannot be applied.
2. Check for Continuity: Verify that the function \(a_n\) is continuous on the interval \([1, \infty)\). If \(a_n\) is discontinuous, the integral test is not applicable.
3. Check for Decreasing Function: Confirm that the function \(a_n\) is decreasing on the interval \([1, \infty)\). This means that for all \(n\), \(a_n \geq a_{n+1}\). If \(a_n\) is not decreasing, the integral test cannot be applied.
4. Evaluate the Integral: Once you have established that the function \(a_n\) meets the above conditions, evaluate the improper integral \(\int_1^{\infty} a_n \, dn\). If the integral converges (i.e., has a finite value), then the series \(\sum_{n=1}^{\infty} a_n\) also converges. Conversely, if the integral diverges (i.e., has an infinite value), then the series \(\sum_{n=1}^{\infty} a_n\) also diverges.
Let’s consider an example to illustrate the application of the integral test:
Suppose we have the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\). To determine if the integral test is applicable, we need to check the following conditions:
1. Positivity: The function \(a_n = \frac{1}{n^2}\) is positive for all \(n\), so this condition is satisfied.
2. Continuity: The function \(a_n\) is continuous on the interval \([1, \infty)\), so this condition is also satisfied.
3. Decreasing Function: The function \(a_n\) is decreasing on the interval \([1, \infty)\), as \(\frac{1}{n^2} \geq \frac{1}{(n+1)^2}\) for all \(n\). Thus, this condition is met.
4. Evaluate the Integral: Now, we evaluate the improper integral \(\int_1^{\infty} \frac{1}{n^2} \, dn\). By using the power rule for integration, we find that the integral converges to \(\frac{\pi^2}{6}\).
Since the integral converges, we can conclude that the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\) also converges.
In summary, to determine if you can apply the integral test to a series, follow the steps outlined above. By ensuring that the function \(a_n\) is positive, continuous, and decreasing, and then evaluating the corresponding improper integral, you can ascertain the convergence or divergence of the series.