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Integral test to show series divergence - Calculus, Mathematics Video Lecture

FAQs on Integral test to show series divergence - Calculus, Mathematics Video Lecture

1. What is the integral test in calculus?
Ans. The integral test is a method used in calculus to determine the convergence or divergence of an infinite series. It involves comparing the series to an improper integral and using the properties of integrals to make conclusions about the series.
2. How does the integral test work to show series divergence?
Ans. The integral test states that if a series ∑an has positive terms and the function f(x) = an is continuous, positive, and decreasing for all x ≥ 1, then the series and the corresponding improper integral ∫f(x)dx both converge or both diverge. Therefore, if the integral diverges, the series will also diverge.
3. Can the integral test be used to determine series convergence?
Ans. Yes, the integral test can be used to determine both convergence and divergence of a series. If the improper integral ∫f(x)dx converges, then the series ∑an also converges. However, if the integral diverges, it does not necessarily mean the series diverges. Additional tests may be needed in such cases.
4. Are there any restrictions on the function f(x) for the integral test to be applicable?
Ans. Yes, there are certain restrictions on the function f(x) for the integral test to be applicable. The function should be continuous, positive, and decreasing for all x ≥ 1. These conditions ensure that the series and the corresponding integral behave similarly and can be used to make conclusions about convergence or divergence.
5. When should the integral test be used to determine series convergence or divergence?
Ans. The integral test is particularly useful when dealing with series that have terms involving functions that are difficult to manipulate or analyze directly. It provides a convenient method to determine the convergence or divergence of such series by comparing them to improper integrals. However, it is important to note that the integral test is not applicable to all series, and other convergence tests may be needed in certain cases.
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