Problem 5
Question
In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\left\\{\frac{n}{n+1}\right\\}\) convergent? If so, what is the limit?
Step-by-Step Solution
Verified Answer
The sequence converges to 1.
1Step 1: Understand the Sequence
We have the sequence \(a_n = \frac{n}{n+1}\). To determine its behavior, we need to find out if this sequence converges to a specific value as \(n\) becomes very large.
2Step 2: Write the Limit Expression
The limit of the sequence \(a_n\) as \(n\) approaches infinity is expressed as \(\lim_{n \to \infty}a_n\). We need to find \(\lim_{n \to \infty} \frac{n}{n+1}\).
3Step 3: Simplify the Expression
Simplify the expression \(\frac{n}{n+1}\) by dividing both the numerator and the denominator by \(n\) (the highest degree term in the denominator):\[ \frac{n}{n+1} = \frac{n/n}{(n+1)/n} = \frac{1}{1 + 1/n} \]
4Step 4: Apply the Limit Laws
Apply the limit laws to \(\lim_{n \to \infty} \frac{1}{1 + 1/n}\). As \(n\) becomes very large, \(\frac{1}{n}\) becomes very small, approaching zero. So,\[ \lim_{n \to \infty} \frac{1}{1 + 1/n} = \frac{1}{1 + 0} = 1 \]
5Step 5: Conclude Convergence
Since the limit exists and is equal to 1, the sequence \(\left\{ \frac{n}{n+1} \right\}\) is convergent. The limit of the sequence is 1.
Key Concepts
Limit of a SequenceReal AnalysisCalculus
Limit of a Sequence
A sequence is a list of numbers, arranged in a specific order. The limit of a sequence tells us where the numbers in the sequence are headed as they extend towards infinity. The limit describes how the values settle around a particular point. In simpler terms, just imagine you're trying to figure out what a sequence looks like way down the line.
To find the limit of the sequence \(\left\{\frac{n}{n+1}\right\}\), we analyze its behavior as \(n\) increases. As \(n\) becomes very large, \(\frac{1}{n}\) approaches zero. When \(n\) is in the hundreds or thousands, \(\frac{1}{n}\) is a number close to zero, like 0.001. This understanding helps simplify the formula \(\frac{n}{n+1}\) to \(\frac{1}{1+1/n}\).
Using limit laws, we recognize that \(\lim_{n \to \infty} \frac{1}{1 + 0} = 1\). This is because as \(n\) grows indefinitely, \(\frac{1}{n}\) gets tinier and essentially vanishes, leaving us with a limit of 1. This means our sequence converges to 1.
To find the limit of the sequence \(\left\{\frac{n}{n+1}\right\}\), we analyze its behavior as \(n\) increases. As \(n\) becomes very large, \(\frac{1}{n}\) approaches zero. When \(n\) is in the hundreds or thousands, \(\frac{1}{n}\) is a number close to zero, like 0.001. This understanding helps simplify the formula \(\frac{n}{n+1}\) to \(\frac{1}{1+1/n}\).
Using limit laws, we recognize that \(\lim_{n \to \infty} \frac{1}{1 + 0} = 1\). This is because as \(n\) grows indefinitely, \(\frac{1}{n}\) gets tinier and essentially vanishes, leaving us with a limit of 1. This means our sequence converges to 1.
Real Analysis
Real analysis is the branch of mathematics that deals with the real numbers and the analytical properties of real-valued functions and sequences. It is all about understanding the deeper properties of calculus and sequences, including concepts like limits, continuity, and convergence.
Within real analysis, one of the core ideas is understanding when a sequence converges. In our example, the sequence \(\left\{\frac{n}{n+1}\right\}\) converges to 1. Real analysis helps us justify this conclusion through proving the rigorous mathematical behavior of numbers. It ensures we are not just observing but proving the sequence approaches a single point (here, the number 1) as \(n\) gets indefinitely large.
Through sophisticated methods like \(\epsilon-\delta\) proofs, real analysis provides strong tools to carefully verify limit behavior. Thus, when analyzing our sequence, real analysis principles dictate why and how the sequence converges.
Within real analysis, one of the core ideas is understanding when a sequence converges. In our example, the sequence \(\left\{\frac{n}{n+1}\right\}\) converges to 1. Real analysis helps us justify this conclusion through proving the rigorous mathematical behavior of numbers. It ensures we are not just observing but proving the sequence approaches a single point (here, the number 1) as \(n\) gets indefinitely large.
Through sophisticated methods like \(\epsilon-\delta\) proofs, real analysis provides strong tools to carefully verify limit behavior. Thus, when analyzing our sequence, real analysis principles dictate why and how the sequence converges.
Calculus
Calculus is the field of mathematics focusing on change and motion. It's largely built around the concepts of differentiation and integration. However, it also provides powerful tools for analyzing sequences and series through limits.
In calculus, the concept of a limit is central. Calculus uses limits to define a lot of its functionality — for example, determining the slope of a curve at a point (differentiation) or the area under a curve (integration).
Finding the limit of a sequence like \(\frac{n}{n+1}\) is a way calculus helps us understand long-term behavior of mathematical functions. As shown, simplifying the sequence's formula helps us apply limits effectively, revealing that as \(n\) goes on to infinity, \(\frac{n}{n+1}\) becomes closer and closer to 1. This makes calculus a powerful tool in real analysis for dissecting specific functions or sequences and understanding precisely where they converge.
In calculus, the concept of a limit is central. Calculus uses limits to define a lot of its functionality — for example, determining the slope of a curve at a point (differentiation) or the area under a curve (integration).
Finding the limit of a sequence like \(\frac{n}{n+1}\) is a way calculus helps us understand long-term behavior of mathematical functions. As shown, simplifying the sequence's formula helps us apply limits effectively, revealing that as \(n\) goes on to infinity, \(\frac{n}{n+1}\) becomes closer and closer to 1. This makes calculus a powerful tool in real analysis for dissecting specific functions or sequences and understanding precisely where they converge.
Other exercises in this chapter
Problem 5
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