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Chapter 9: Problem 3

Can the Integral Test be used to determine whether a series diverges?

Short Answer

Expert verified
Answer: Yes, the Integral Test can be used as an indication of whether a series diverges. However, it is more conclusive in determining convergence. If the improper integral from 1 to 鈭� results in an infinite value or doesn't converge, it suggests that the series is likely to diverge, but it may not always guarantee the divergence.

Step by step solution

01

Understanding the Integral Test

The Integral Test is a comparison test that can be used to determine the convergence or divergence of a given series. It states that if f(x) is a continuous, positive, and decreasing function on the interval [1, 鈭�), and 鈭�(f(n)) is a series, then the series 鈭�(f(n)) converges if and only if the improper integral, 鈭�(f(x)dx), converges from 1 to 鈭�.
02

Using the Integral Test for Convergence

To determine if a series converges using the Integral Test, integrate the function from 1 to 鈭�. If the result is a finite value, the series converges. Otherwise, it might indicate that the series diverges.
03

Using the Integral Test for Divergence

As seen in the previous step, if the improper integral results in an infinite value or does not converge, it can indicate that the series diverges. However, it should also be noted that the Integral Test only provides conclusive evidence when it comes to convergence. If the improper integral diverges, it is a strong indication of divergence but doesn't always guarantee divergence for the original series.
04

Checking if the Integral Test can Determine Divergence

The Integral Test can indeed be used as an indication of whether a series diverges. If the improper integral from 1 to 鈭� results in an infinite value or doesn't converge, then it's a strong indication that the series diverges. However, it's important to note that the Integral Test only provides definitive evidence for convergence. In the case of divergence, the Integral Test can be viewed as a strong indicator, but it may not always guarantee the divergence.

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Most popular questions from this chapter

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