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Chapter 9: Problem 4

What is the first test you should use in analyzing the convergence of a series?

Short Answer

Expert verified
Answer: The first test to use when analyzing the convergence of a series is the Divergence Test, or the nth-term test. To apply this test, follow these steps: 1. Determine the series 鈭慳_n that you want to analyze. 2. Calculate the limit of the terms a_n as n approaches infinity. 3. If the limit is not equal to zero, the series diverges. If the limit is equal to zero, the test is inconclusive, and you'll need to apply other convergence tests to determine the behavior of the series.

Step by step solution

01

Understanding the Divergence Test

The Divergence Test, or the nth-term test, is a basic test to check the convergence of a given series. The test states that if the limit of the terms in the series as n approaches infinity is not equal to zero, the series diverges. That is, given a series 鈭慳_n: If lim(n鈫掆垶) a_n 鈮� 0, then the series 鈭慳_n diverges. However, if the limit is equal to zero: If lim(n鈫掆垶) a_n = 0, the test is inconclusive, and we'll have to use other convergence tests to determine if the series converges or not.
02

Applying the Divergence Test

To use the Divergence Test, follow these steps: 1. Determine the series 鈭慳_n that you want to analyze. 2. Calculate the limit of the terms a_n as n approaches infinity. 3. If the limit is not equal to zero, the series diverges. If the limit is equal to zero, the test is inconclusive, and you'll need to apply other convergence tests to determine the behavior of the series. In conclusion, the first test to use when analyzing the convergence of a series is the Divergence Test, or the nth-term test. This test can help to identify if a series definitely diverges, but it can't guarantee convergence, so further tests may be required.

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Most popular questions from this chapter

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=2 a_{n}\left(1-a_{n}\right) ; a_{0}=0.3, n=0,1,2, \dots$$

For a positive real number \(p,\) how do you interpret \(p^{p^{p \cdot *}},\) where the tower of exponents continues indefinitely? As it stands, the expression is ambiguous. The tower could be built from the top or from the bottom; that is, it could be evaluated by the recurrence relations \(a_{n+1}=p^{a_{n}}\) (building from the bottom) or \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{0}=p\) in either case. The two recurrence relations have very different behaviors that depend on the value of \(p\). a. Use computations with various values of \(p > 0\) to find the values of \(p\) such that the sequence defined by (2) has a limit. Estimate the maximum value of \(p\) for which the sequence has a limit. b. Show that the sequence defined by (1) has a limit for certain values of \(p\). Make a table showing the approximate value of the tower for various values of \(p .\) Estimate the maximum value of \(p\) for which the sequence has a limit.

Given any infinite series \(\Sigma a_{k},\) let \(N(r)\) be the number of terms of the series that must be summed to guarantee that the remainder is less than \(10^{-r}\), where \(r\) is a positive integer. a. Graph the function \(N(r)\) for the three alternating \(p\) -series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{p}},\) for \(p=1,2,\) and \(3 .\) Compare the three graphs and discuss what they mean about the rates of convergence of the three series. b. Carry out the procedure of part (a) for the series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}\) and compare the rates of convergence of all four series.

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\) ). a. \(\sum_{k=0}^{\infty} 0.72^{k}\) b. \(\sum_{k=0}^{\infty}(-0.25)^{k}\)

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. A material transmutes \(50 \%\) of its mass to another element every 10 years due to radioactive decay. Let \(M_{n}\) be the mass of the radioactive material at the end of the \(n\) th decade, where the initial mass of the material is \(M_{0}=20 \mathrm{g}.\)
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