When Does the Integral Test Not Apply?

The integral test is a useful tool for determining the convergence or divergence of an infinite series. However, there are certain conditions under which the integral test does not apply. Here are the main scenarios:

1. Non-Positive Terms: The integral test requires that the terms of the series are positive. If the series has negative terms or alternates between positive and negative, the integral test cannot be applied.
2. Non-Continuous Function: The function corresponding to the series must be continuous on the interval [1, ∞). If the function has discontinuities in this interval, the integral test is not applicable.
3. Non-Decreasing Function: The function must be decreasing on the interval [1, ∞). If the function is not decreasing, the integral test cannot be used to determine the convergence or divergence of the series.
4. Non-Integrable Function: The function must be integrable on the interval [1, ∞). If the function is not integrable, the integral test is not applicable.

In summary, the integral test does not apply if the series has non-positive terms, if the corresponding function is not continuous, not decreasing, or not integrable on the interval [1, ∞).