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Your textbook must have a theorem in which they present the integral test. Find it and see what it says.InvalidID said:
Integrating from 1 to ∞ allows us to capture the behavior of the function as x approaches infinity. This is important because it helps us determine if the series converges or diverges.
Yes, in some cases, a different lower limit can be used. However, 1 is often the most convenient lower limit to use because it simplifies the calculations and allows us to easily compare the series with the p-series test.
The Integral Test can be used when the series is positive, continuous, and decreasing. If these conditions are met, then we can use the Integral Test to determine the convergence or divergence of the series.
If the integral of the function from 1 to ∞ converges, then the series also converges. If the integral diverges, then the series also diverges. This is known as the Cauchy Condensation Test.
No, the Integral Test can only be used for series with positive terms. It cannot be used for alternating series or series with negative terms. In those cases, other tests such as the Alternating Series Test or the Comparison Test should be used instead.
