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Chapter 9: Problem 81

The terms of a series \(\sum_{n=1}^{\infty} a_{n}\) are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. $$ a_{1}=\frac{1}{3}, a_{n+1}=\left(1+\frac{1}{n}\right) a_{n} $$

Short Answer

Expert verified
The series \(\sum_{n=1}^{\infty} a_{n}\), where \(a_{n+1}=\left(1+\frac{1}{n}\right) a_{n}\) and \(a_{1}=\frac{1}{3}\) is indeterminate regarding its convergence or divergence based on the Ratio Test. Additional analysis or tests would be required for a definitive conclusion.

Step by step solution

01

Observation

Observe that the sequence \(a_{n}\) is positive and notice that the terms are increasing with n. The next \(a_{n+1}\) is directly proportional to \(a_{n}\) and each subsequent term becomes larger.
02

Apply Ratio Test

Apply the Ratio Test (D'Alembert), which is a popular method for analyzing convergence of series with positive terms. We form the ratio of \(a_{n+1}\) to \(a_{n}\) and see how this ratio behaves as \(n\) tends to infinity.
03

Calculation

\[ \frac{a_{n+1}}{a_{n}}= \left(1+\frac{1}{n}\right) \]. The limit as \(n\) approaches infinity of this ratio is \(1\).
04

Conclusion

If the ratio test yields 1, it is inconclusive and does not provide any information about the behavior of the series. Therefore, we cannot make a conclusion about the convergence or divergence of the series solely based on this ratio test.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Recursive Series
In mathematics, a recursive series is a sequence of numbers where each subsequent term is defined based on the previous ones, often through a specific formula. In this problem, the sequence is given as \(a_1 = \frac{1}{3}\) and \(a_{n+1} = \left(1+\frac{1}{n}\right) a_n\). This means each term is created by multiplying the current term by a factor involving \(n\).

Recursive series are interesting because they build complexity over time. The terms of a recursive series can either grow larger, shrink, or stabilize depending on the recursive rule. Understanding the pattern helps in predicting whether the series will converge to a limit or diverge towards infinity. Recognizing the behavior of recursive sequences is crucial in determining their overall behavior.
Ratio Test
The Ratio Test is a powerful tool for assessing the convergence of an infinite series. It's particularly effective for series with positive terms, like the one described here. The idea is to look at the ratio \( \frac{a_{n+1}}{a_n} \) and determine its limit as \( n \) approaches infinity.

In our problem, this ratio \( \frac{a_{n+1}}{a_n} = \left(1+\frac{1}{n}\right) \) simplifies to approach 1 when \( n \to \infty \). If this limit is less than 1, the series converges; if greater than 1, it diverges. However, when the limit is exactly 1, the test is inconclusive. This means that additional methods or tests must be employed to determine the series' behavior.
  • Gives a quick insight into the series convergence.
  • For limits less than 1, convergence is assured.
  • Limits greater than 1 indicate divergence.
  • Exact limit of 1 necessitates further analysis.
Convergence and Divergence
Convergence and divergence describe the behavior of an infinite series as the number of terms grows. A series converges if the sum of its terms approaches a finite number. On the other hand, it diverges if the sum increases indefinitely or oscillates.

Identifying convergence or divergence helps mathematicians understand whether a series has a meaningful total value or not. In this exercise, since the ratio test was inconclusive, we must look deeper or use other tools like the root test or comparison test to determine the series' overall behavior.
  • Convergence implies a finite sum result.
  • Divergence means the series does not sum to a finite value.
  • Inconclusive tests require alternative approaches.
  • This understanding aids in many applied mathematical fields.
Infinite Series Analysis
Analyzing infinite series can seem daunting due to their endless nature. The primary goal is to understand the series' general behavior as the number of terms increases. This involves looking at patterns, behavior, and whether or not the series converges to a finite value.

In the given recursive series, the attempt to apply the ratio test provided no definitive answer, suggesting more in-depth analysis or alternative methods are necessary. Understanding when and how to apply certain tests like the ratio test, root test, or p-series test is vital in infinite series analysis.
  • Essential for complex mathematical modeling.
  • Requires examining both initial and long-term behaviors.
  • Combines different tests for a comprehensive analysis.
  • Helps predict future behavior of the series.

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Most popular questions from this chapter

Multiplier Effect The annual spending by tourists in a resort city is \(\$ 100\) million. Approximately \(75 \%\) of that revenue is again spent in the resort city, and of that amount approximately \(75 \%\) is again spent in the same city, and so on. Write the geometric series that gives the total amount of spending generated by the \(\$ 100\) million and find the sum of the series.

Find two divergent series \(\sum a_{n}\) and \(\sum b_{n}\) such that \(\Sigma\left(a_{n}+b_{n}\right)\) converges.

Fibonacci Sequence In a study of the progeny of rabbits, Fibonacci (ca. \(1170-\) ca. 1240 ) encountered the sequence now bearing his name. It is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}, \quad\) where \(\quad a_{1}=1\) and \(a_{2}=1\). (a) Write the first 12 terms of the sequence. (b) Write the first 10 terms of the sequence defined by $$ b_{n}=\frac{a_{n+1}}{a_{n}}, \quad n \geq 1 . $$ (c) Using the definition in part (b), show that $$ b_{n}=1+\frac{1}{b_{n-1}} $$ (d) The golden ratio \(\rho\) can be defined by \(\lim _{n \rightarrow \infty} b_{n}=\rho\). Show that \(\rho=1+1 / \rho\) and solve this equation for \(\rho\).

Use the formula for the \(n\) th partial sum of a geometric series \(\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}\) Salary You go to work at a company that pays \(\$ 0.01\) for the first day, \(\$ 0.02\) for the second day, \(\$ 0.04\) for the third day, and so on. If the daily wage keeps doubling, what would your total income be for working (a) 29 days, (b) 30 days, and (c) 31 days?

The random variable \(n\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\) $$ P(n)=\frac{1}{3}\left(\frac{2}{3}\right)^{n} $$
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