Which of the sequences {an} in Exercises 27-90 converge, and which diverge? Find the limit of ea

step 1 We're given a sequence, and we're asked to determine if the sequence converges and diverges, and

Which of the sequences {an} in Exercises 27-90 converge, and...

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Key Concepts

Improper Integrals

Improper integrals involve integration over an unbounded interval or where the integrand has unbounded behavior within the interval. In problems like this, the integral is evaluated as the limit of a definite integral as the interval’s endpoint approaches infinity, which is essential for determining whether the overall expression converges or diverges.

p-Test for Convergence

The p-test is a specific criterion used to assess the convergence of integrals of the form ??? (1/x?) dx. According to this test, the integral converges if and only if the exponent p is greater than 1, and diverges when p is less than or equal to 1. This test is fundamental when dealing with sequences or series defined via such integrals.

Evaluating Limits of Sequences Defined by Integrals

For sequences defined as integrals with a variable upper limit, convergence is found by first computing the antiderivative and then taking the limit as the upper limit approaches infinity. If the resulting expression yields a finite number, the sequence converges to that limit; otherwise, it is divergent. In the given context, with p > 1, the evaluation leads to a finite limit.

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