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Chapter 9: Problem 45

Recursively Defined Terms Which of the series \(\Sigma_{n=1}^{\infty} a_{n}\) defined by the formulas in Exercises \(45-54\) converge, and which diverge? Give reasons for your answers. $$a_{1}=2, \quad a_{n+1}=\frac{1+\sin n}{n} a_{n}$$

Short Answer

Expert verified
The series converges due to the term limit ratio approaching zero.

Step by step solution

01

Understanding the Initial Term

The series starts with the first term given by \(a_1 = 2\). This will be the anchor for determining the nature of the series as each subsequent term is calculated based on this.
02

Defining the Recursive Formula

The recursive formula is given by \(a_{n+1} = \frac{1 + \sin n}{n} a_n\). This allows us to express each term in the series recursively based on the previous term \(a_n\).
03

Analyzing the Term Ratio

Calculate \(\frac{a_{n+1}}{a_n}\) which simplifies to \(\frac{1+\sin n}{n}\). Since \(\sin n\) oscillates between \(-1\) and \(1\), the expression \(\frac{1+\sin n}{n}\) will have a range of values from \(\frac{0}{n}\) to \(\frac{2}{n}\).
04

Limit Exploration

As \(n\) goes to infinity, \(\frac{1+\sin n}{n}\) approaches \(0\) because \(1+\sin n\) is bounded between \(0\) and \(2\), resulting in terms becoming negligible. This suggests that the series might converge as subsequent terms reduce.
05

Applying the Ratio Test

The ratio test involves examining the absolute value of \(\frac{a_{n+1}}{a_n}\). Simplifying, because \(\lim_{n \to \infty} \frac{1+\sin n}{n} = 0\), the ratio test suggests convergence because \(r < 1\), where \(r\) is the limit of the term ratio.
06

Conclusion on Convergence

The limit of the term ratio being zero, such that subsequent terms decay rapidly, indicates that the series \(\Sigma_{n=1}^{\infty} a_{n}\) converges. The presence of \(n\) in the denominator increases more rapidly than the oscillation of \(\sin n\), enough for the terms to shrink and converge.

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Key Concepts

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

Recursive Sequences
A recursive sequence is a sequence in which each term is defined based on previous terms using a specific rule or formula. In our given problem, we have a sequence where each term is calculated using the formula \(a_{n+1} = \frac{1 + \sin n}{n} a_n\). This means that to find any term, you need to know the term immediately before it and apply the function dictated by the recursive formula.

Here are some key points to understand recursive sequences:
  • Initial Condition: Begins with \(a_1 = 2\), setting the starting point for the sequence.

  • Recursion Formula: Each subsequent term\(a_{n+1}\) relies on \(a_n\), maintaining a defined relationship throughout the sequence.

  • Behavior: Recursive sequences can exhibit behavior such as cyclic, convergent, or divergent depending on their recursive relation.
Understanding recursive sequences is crucial for predicting the sequence behavior as \(n\) increases, especially when determining convergence or divergence.
Ratio Test
The ratio test is a method used to determine whether a series converges or diverges by examining the limit of the absolute value of the ratio of successive terms. In simple terms, it tells us whether the terms in the series are getting smaller fast enough for the series to have a finite sum.

In our case, we have the ratio:
  • Calculate the ratio: \(\frac{a_{n+1}}{a_n} = \frac{1+\sin n}{n}\).

  • Observe that the sine function \(\sin n\) oscillates between \(-1\) and \(1\), so \(1 + \sin n\) oscillates between \(0\) and \(2\).

  • Analyze Convergence: As \(n\) becomes large, \(\frac{1+\sin n}{n}\) approaches zero, suggesting convergence based on the ratio test. This meets the criterion of the ratio test, where the limit \(L < 1\), ensuring convergence.
By analyzing these elements, the ratio test helps confirm that the series converges since the term ratio diminishes as the number of terms increases.
Limit Analysis
Limit analysis is the study of what happens to the terms of a sequence or series as they progress towards infinity. It helps in understanding whether a series converges or diverges.

In the context of the given recursive sequence:
  • Bound Analysis: Since \(1+\sin n\) is always between \(0\) and \(2\), \(\frac{1+\sin n}{n}\) clearly approaches \(0\) as \(n\) increases, confirming that the contribution of each term to the series sum decreases.

  • Infinite Limit: As \(n\) approaches infinity, observing this limit confirms that the sequence terms become negligible, encouraging series convergence.

  • Convergence Insight: With \(\lim_{n \to \infty} \frac{1+\sin n}{n} = 0\), terms shrink progressively, reducing their impact on the accumulation of the series as a whole and leading to the conclusion drawn from a convergence perspective.
Thus, limit analysis plays a vital role in predicting and validating the behavior of an infinite series, revealing how terms diminish and ultimately supporting convergence.

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Most popular questions from this chapter

Prove that \(\lim _{n \rightarrow \infty} x^{1 / n}=1,(x>0)\).

Which of the sequences converge, and which diverge? Give reasons for your answers. $$a_{1}=1, \quad a_{n+1}=2 a_{n}-3$$

Uniqueness of least upper bounds Show that if \(M_{1}\) and \(M_{2}\) are least upper bounds for the sequence \(\left\\{a_{n}\right\\},\) then \(M_{1}=M_{2}\) That is, a sequence cannot have two different least upper bounds.

It is not yet known whether the series $$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \sin ^{2} n} $$converges or diverges. Use a CAS to explore the behavior of the series by performing the following steps. a. Define the sequence of partial sums $$ s_{k}=\sum_{n=1}^{k} \frac{1}{n^{3} \sin ^{2} n} $$ What happens when you try to find the limit of \(s_{k}\) as \(k \rightarrow \infty ?\) Does your CAS find a closed form answer for this limit? b. Plot the first 100 points \(\left(k, s_{k}\right)\) for the sequence of partial sums. Do they appear to converge? What would you estimate the limit to be? c. Next plot the first 200 points \(\left(k, s_{k}\right) .\) Discuss the behavior in your own words. d. Plot the first 400 points \(\left(k, s_{k}\right) .\) What happens when \(k=355 ?\) Calculate the number \(355 / 113 .\) Explain from you calculation what happened at \(k=355 .\) For what values of \(k\) would you guess this behavior might occur again?

$$\text { Let } a_{n}=\left\\{\begin{array}{ll} n / 2^{n}, & \text { if } n \text { is a prime number } \\ 1 / 2^{n}, & \text { otherwise. } \end{array}\right.$$ Does \(\Sigma a_{n}\) converge? Give reasons for your answer.
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