Problem 10 Find the limit of the sequence \... [FREE SOLUTION]

Chapter 10: Problem 10

Find the limit of the sequence $\left\\{a_{n}\right\\}$ if $1-\frac{1}{n} < a_{n} < 1+\frac{1}{n},$ for every integer $n \geq 1$

Short Answer

Expert verified

Answer: The limit of the given sequence ${a_n}$ is 1.

Step by step solution

Recognize the type of problem

This is a problem of finding the limit of a sequence. To solve this, let's use the given information about the inequality that bounds the sequence ${a_n}$.

Find the limit of the lower bounding sequence

Let's first determine the limit of the lower bounding sequence, given by $1-\frac{1}{n}$. For this, calculate the limit as $n \to \infty$. As n tends to infinity: $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right) = 1 - \lim_{n\to\infty}\frac{1}{n}$$. Since the limit of the sequence $\frac{1}{n}$ when $n$ tends to infinity is 0, we have: $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right) = 1 - 0 = 1$$.

Find the limit of the upper bounding sequence

Next, we need to find the limit of the upper bounding sequence, given by $1+\frac{1}{n}$. For this, we again calculate the limit as $n \to \infty$. As n tends to infinity: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right) = 1 + \lim_{n\to\infty}\frac{1}{n}$$. Similar to the previous step, since the limit of the sequence $\frac{1}{n}$ when $n$ tends to infinity is 0, we have: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right) = 1 + 0 = 1$$.

Apply the Squeeze Theorem

Now we have the limits of both lower and upper bounding sequences: $$\lim_{n\to\infty}\left(1-\frac{1}{n}\right) = \lim_{n\to\infty}\left(1+\frac{1}{n}\right) = 1$$. Since the inequality is true for every integer $n \geq 1$: $$1-\frac{1}{n} \leq a_{n} \leq 1+\frac{1}{n}$$, We can apply the squeeze theorem (or sandwich theorem) which states that if a sequence is bounded by two sequences that have the same limit, then the sequence in between must have the same limit as well: $$\lim_{n\to\infty}a_{n} = 1$$. Therefore, the limit of the given sequence ${a_n}$ is 1.

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Key Concepts

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

Squeeze Theorem

The Squeeze Theorem, also known as the Sandwich Theorem, is a powerful tool in calculus for finding limits. It is used when a sequence or function is squeezed between two other functions or sequences whose limits are known and identical.Imagine you have three sequences:

One sequence that's always greater (an upper bound).
One sequence that's always smaller (a lower bound).
The sequence you're interested in finding the limit for, sitting right in between.

If the upper and lower bounding sequences converge to the same limit as they approach infinity, the squeezed sequence must converge to the same limit.
In our example, the sequence $a_n$ is squeezed between $1-\frac{1}{n}$ and $1+\frac{1}{n}$, both of which converge to 1 as $n\to\infty$. Hence, by the Squeeze Theorem, $a_n$ also converges to 1.

Upper and Lower Bounds

Understanding upper and lower bounds is essential for working with sequences and their limits. An upper bound is a value that a sequence never exceeds, and a lower bound is a value that a sequence is never less than.For a sequence to be bounded:

There must exist a value that is an upper bound for all elements of the sequence.
There must also exist a value that is a lower bound for all elements of the sequence.

In the given exercise, the sequence $a_n$ is bounded by $1-\frac{1}{n}$ (lower bound) and $1+\frac{1}{n}$ (upper bound). As $n\to\infty$, these bounds approach 1, providing a controlled environment to apply the Squeeze Theorem effectively.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index (usually denoted $n$) becomes very large. Finding this limit is a fundamental aspect of calculus and analysis.To determine the limit of a sequence:

Consider the behavior of the function as $n\to\infty$.
Identify any patterns or recurrent relations in the terms.
Apply theorems like the Squeeze Theorem, especially when dealing with bounded sequences.

In the exercise, both the lower and upper bounds of $a_n$ converge to the number 1. This means that $a_n$ must also converge to 1, giving us the limit $\lim_{n\to\infty}a_n = 1$. Understanding this concept helps you analyze and predict the behavior of sequences effectively.

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91影视

Find the limit of the sequence \(\left\\{a_{n}\right\\}\) if \(1-\frac{1}{n} < a_{n} < 1+\frac{1}{n},\) for every integer \(n \geq 1\)

Short Answer

Step by step solution

Recognize the type of problem

Find the limit of the lower bounding sequence

Find the limit of the upper bounding sequence

Apply the Squeeze Theorem

Key Concepts

Squeeze Theorem

Upper and Lower Bounds

Limit of a Sequence

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