Problem 7 In the following problems, find ... [FREE SOLUTION]

Chapter 1: Problem 7

In the following problems, find the limit of the given sequence as $n \rightarrow \infty$. $$\left(1+n^{2}\right)^{1 / \ln n}$$

Short Answer

Expert verified

The limit of the sequence is $e^2$.

Step by step solution

Determine the Form of the Sequence as n Approaches Infinity

Examine the sequence $\left(1+n^{2}\right)^{1 / \ln n}$. As $n$ approaches infinity, both $n^{2}$ and $\ln n$ also approach infinity. We need to simplify this expression for large $n$.

Use Logarithms to Simplify the Calculation

Take the natural logarithm of the sequence to make the limit easier to handle. Let $a_n = \left(1+n^2\right)^{1/ \ln n}$. Then consider $\ln(a_n) = \frac{\ln(1+n^{2})}{\ln n}$.

Apply L'H么pital's Rule if Necessary

To find the limit of $\frac{\ln(1+n^{2})}{\ln n}$, observe that as $n \rightarrow \infty$, the form is 鈭�/鈭�. Apply L'H么pital's rule by differentiating the numerator and the denominator with respect to $n$.

Differentiate Numerator and Denominator

Differentiate $\ln(1+n^{2})$ and $\ln n$ with respect to $n$. The derivative of $\ln(1+n^{2})$ is $\frac{2n}{1+n^{2}}$, and the derivative of $\ln n$ is $\frac{1}{n}$. Thus, the limit becomes:\[\lim_{n \to \infty} \frac{2n/(1+n^{2})}{1/n} = \lim_{n \to \infty} \frac{2n^2}{1+n^{2}} = \lim_{n \to \infty} \frac{2}{1/n^{2} + 1} = 2.\]

Conclude the Limit for the Original Sequence

Since $\ln(a_n) = 2 $, we have that $a_n = e^{\ln(a_n)} = e^{2}.$ Hence,$a_n = \left(1+n^{2}\right)^{1/ \ln n}$ approaches $e^2$ as $n \rightarrow \infty.$

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Key Concepts

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

Limit of a Sequence

The limit of a sequence helps us understand the behavior of the sequence as the index (usually denoted by 'n') approaches infinity. For a sequence $a_n$, if as $n \rightarrow \infty$, the terms of the sequence approach a certain value L, we denote it as \lim_{n \to \infty} a_n = L\. For example, finding the limit of the sequence $\left(1+n^{2}\right)^{1 / \ln n}$, we are determining what this sequence approaches as $n$ gets very large. This concept allows us to simplify complex expressions and is a core idea in calculus.

Natural Logarithm

The natural logarithm, denoted as \ln\, is a logarithm to the base \e\, where \e\ is an irrational constant approximately equal to 2.71828. It is often used in calculus to simplify expressions, especially when dealing with exponential growth or decay. By taking the natural logarithm of both sides of an equation, we can convert multiplicative relationships into additive ones, making them easier to manipulate. In our problem, taking the natural logarithm of the sequence $\left(1+n^{2}\right)^{1 / \ln n}$ helped simplify the limit calculation.

L'H么pital's Rule

L'H么pital's rule is a powerful tool for handling indeterminate forms like \frac{0}{0}\ or \frac{\infty}{\infty}\. It states that if two functions are differentiable and their limit results in an indeterminate form, we can take the derivatives of the numerator and the denominator to evaluate the limit. In our example, to find the limit of \frac{\ln(1+n^{2})}{\ln n}\ as $n \rightarrow \infty$, we applied L'H么pital's rule. By differentiating both \ln(1+n^{2})\ and \ln n\ with respect to $n$, the limit became easier to solve.

Differentiation

Differentiation involves finding the derivative of a function. The derivative measures how a function changes as its input changes. For instance, the derivative of \ln n\ with respect to $n$ is \frac{1}{n}\, and the derivative of \ln(1+n^{2})\ with respect to $n$ is \frac{2n}{1+n^{2}}\. By determining these derivatives, we could apply L'H么pital's rule to solve our limit problem. Differentiation is fundamental in calculus for analyzing and interpreting the behavior of functions.

Infinity

Infinity, denoted by the symbol \infty\, represents an unbounded quantity that's larger than any real number. In calculus, we frequently deal with limits that involve infinity to analyze the behavior of sequences and functions as they grow without bound. For the sequence $\left(1+n^{2}\right)^{1 / \ln n}$, we sought the limit as $n \rightarrow \infty$. Understanding how expressions behave as they approach infinity is crucial for solving a wide range of mathematical problems.

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91影视

In the following problems, find the limit of the given sequence as \(n \rightarrow \infty\). $$\left(1+n^{2}\right)^{1 / \ln n}$$