Convergence Of The Sequence 3/4, 4/5, 5/6, 6/7

In this article, we will delve into the fascinating world of sequences and their convergence. Specifically, we will analyze the sequence 3/4, 4/5, 5/6, 6/7, ..., and determine its behavior as it progresses infinitely. Our primary goal is to identify whether this sequence converges, and if so, to what limit it converges. This exploration will involve a step-by-step examination of the sequence's terms, the identification of a general formula, and a rigorous analysis of its limiting behavior. The question at hand is: which statement accurately describes this sequence? Does it diverge, converge to 1, converge to infinity, or converge to some other value? By the end of this discussion, you will have a clear understanding of how to approach such problems and the mathematical principles underlying the concept of convergence.

Identifying the Pattern and General Term

To begin our analysis, let's first examine the given sequence: 3/4, 4/5, 5/6, 6/7, .... A crucial first step in understanding any sequence is to identify the pattern governing its terms. In this case, we observe that the numerator of each fraction is one less than the denominator. Moreover, the numerators increase sequentially, as do the denominators. This pattern suggests that we can express the general term of the sequence, often denoted as a_n, using a simple algebraic expression. The general term allows us to represent any term in the sequence without having to list out all the preceding terms. It provides a concise formula that captures the essence of the sequence's behavior. By carefully observing the relationship between the term number (n) and the corresponding fraction, we can derive this general term. This is a fundamental step in analyzing the sequence's long-term behavior and determining its convergence or divergence. The ability to identify patterns and express them mathematically is a cornerstone of mathematical analysis, and this example provides a practical illustration of this skill.

Let's break down how we arrive at the general term. The first term (n=1) is 3/4, the second term (n=2) is 4/5, the third term (n=3) is 5/6, and so on. Notice that the numerator is always 'n+2' and the denominator is 'n+3'. Therefore, the general term a_n can be expressed as (n+2)/(n+3). This expression is the key to understanding the sequence's behavior as n approaches infinity. Now that we have a general formula, we can use it to analyze the limiting behavior of the sequence.

Analyzing the Limiting Behavior

Now that we have identified the general term of the sequence as a_n = (n+2)/(n+3), we can proceed to analyze its limiting behavior. Limiting behavior refers to what happens to the terms of the sequence as n becomes infinitely large. In other words, we want to find the limit of a_n as n approaches infinity. This is a crucial step in determining whether the sequence converges or diverges. If the limit exists and is a finite number, the sequence converges to that limit. If the limit does not exist or is infinite, the sequence diverges. The concept of a limit is fundamental to calculus and mathematical analysis, and it provides a rigorous way to define the notion of a sequence approaching a specific value.

To find the limit, we can use several techniques. One common method is to divide both the numerator and the denominator by the highest power of n present in the expression. In this case, the highest power of n is simply n. So, we divide both the numerator (n+2) and the denominator (n+3) by n. This gives us:

a_n = (n+2)/(n+3) = (n/n + 2/n) / (n/n + 3/n) = (1 + 2/n) / (1 + 3/n)

Now, as n approaches infinity, the terms 2/n and 3/n approach zero. This is because dividing a constant by an increasingly large number results in a value that gets closer and closer to zero. Therefore, the expression (1 + 2/n) approaches 1, and the expression (1 + 3/n) also approaches 1. Consequently, the limit of a_n as n approaches infinity is:

lim (n→∞) a_n = lim (n→∞) (1 + 2/n) / (1 + 3/n) = 1/1 = 1

This result tells us that the sequence converges to 1. As we move further and further along the sequence, the terms get closer and closer to the value 1. This understanding of limiting behavior is essential for various applications in mathematics, physics, and engineering.

Evaluating the Options

Now that we have determined that the sequence converges to 1, let's evaluate the given options and identify the correct statement. The options were:

A. The sequence diverges. B. The sequence converges to 1. C. The sequence converges to ∞. D. The sequence converges to.

Based on our analysis, we have conclusively shown that the sequence converges to 1. Therefore, option B is the correct statement. Option A is incorrect because we have established that the sequence converges, not diverges. Option C is also incorrect because the sequence converges to a finite value (1), not infinity. Option D is incomplete and doesn't provide a specific limit, so it cannot be the correct answer. This process of elimination, combined with our rigorous mathematical analysis, allows us to confidently select the correct statement.

It is important to note that understanding the difference between convergence and divergence is crucial in mathematics. A convergent sequence approaches a specific finite value, while a divergent sequence does not. Divergent sequences can either oscillate without approaching a limit or increase (or decrease) without bound. The concept of convergence is fundamental to many areas of mathematics, including calculus, real analysis, and numerical analysis.

Conclusion

In conclusion, through a step-by-step analysis, we have successfully determined the behavior of the sequence 3/4, 4/5, 5/6, 6/7, .... We first identified the pattern and expressed the general term as a_n = (n+2)/(n+3). Then, we analyzed the limiting behavior of the sequence by finding the limit of a_n as n approaches infinity. We found that the limit is 1, which means the sequence converges to 1. Finally, we evaluated the given options and confirmed that the correct statement is: The sequence converges to 1.

This exercise demonstrates the power of mathematical analysis in understanding the behavior of sequences. By identifying patterns, expressing them mathematically, and analyzing limiting behavior, we can gain deep insights into the properties of these fundamental mathematical objects. The concepts and techniques discussed in this article are applicable to a wide range of problems in mathematics and related fields. Understanding convergence and divergence is essential for anyone pursuing further studies in mathematics, physics, engineering, or computer science. The ability to analyze sequences and series is a cornerstone of mathematical literacy and a valuable skill for problem-solving in various contexts.

This analysis highlights the importance of careful observation, pattern recognition, and the application of mathematical principles to arrive at a definitive conclusion. The process of understanding the convergence of a sequence involves a combination of algebraic manipulation, limit evaluation, and logical reasoning. By mastering these skills, you can confidently tackle similar problems and gain a deeper appreciation for the beauty and power of mathematics.