Using The Given Information, Find The Value Of:$\[ \lim _{n \rightarrow \infty} E \sqrt[n]{n!} \left[ \log \left(1+\frac{17}{n}\right) - \frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi N})^{\frac{1}{n}}} \right] \\]The Limits

Introduction

In this article, we will delve into the evaluation of a complex limit expression involving exponential, logarithmic, and root functions. The given expression is:

$$\lim _{n \rightarrow \infty} e \sqrt[n]{n!} \left[ \log \left(1+\frac{17}{n}\right) - \frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}} \right]$$

We will break down the expression into manageable parts, apply various limit properties, and utilize Stirling's approximation to simplify the expression and ultimately evaluate the limit.

Breaking Down the Expression

The given expression can be broken down into three main components:

1. Exponential and Root Functions: $e \sqrt[n]{n!}$
2. Logarithmic Function: $\log \left(1+\frac{17}{n}\right)$
3. Fractional Expression: $\frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}}$

We will analyze each component separately and then combine the results to evaluate the overall limit.

Component 1: Exponential and Root Functions

The first component involves the exponential function $e$ multiplied by the $n$th root of $n!$. We can rewrite this as:

$$e \sqrt[n]{n!} = e \left( \frac{n!}{n^n} \right)^{\frac{1}{n}}$$

Using Stirling's approximation, we can simplify the expression as:

$$e \left( \frac{n!}{n^n} \right)^{\frac{1}{n}} \approx e \left( \frac{e^n}{n^n} \right)^{\frac{1}{n}} = e \left( \frac{e}{n} \right) = \frac{e^2}{n}$$

Component 2: Logarithmic Function

The second component involves the logarithmic function $\log \left(1+\frac{17}{n}\right)$. We can rewrite this as:

$$\log \left(1+\frac{17}{n}\right) = \log \left(1 + \frac{17}{n} \right) = \log \left( \frac{n+17}{n} \right)$$

Using the logarithmic property $\log \left( \frac{a}{b} \right) = \log a - \log b$, we can simplify the expression as:

$$\log \left( \frac{n+17}{n} \right) = \log (n+17) - \log n$$

Component 3: Fractional Expression

The third component involves the fractional expression $\frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}}$. We can rewrite this as:

$$\frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}} = \frac{e^{\frac{8}{n}} - 1}{\sqrt{2 \pi n}^{\frac{1}{n}}}$$

Using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$, we can simplify the expression as:

$$\frac{e^{\frac{8}{n}} - 1}{\sqrt{2 \pi n}^{\frac{1}{n}}} = \frac{e^{\frac{8}{n}} - 1}{\sqrt{2 \pi n}^{\frac{1}{n}}} \cdot \frac{e^{-\frac{8}{n}}}{e^{-\frac{8}{n}}} = \frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{\sqrt{2 \pi n}^{\frac{1}{n}}}$$

Using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$, we can simplify the expression as:

$$\frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{\sqrt{2 \pi n}^{\frac{1}{n}}} = \frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{\sqrt{2 \pi n}^{\frac{1}{n}}} \cdot \frac{e^{\frac{8}{n}}}{e^{\frac{8}{n}}} = \frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}}$$

Using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$, we can simplify the expression as:

$$\frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} = \frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} \cdot \frac{e^{-\frac{8}{n}}}{e^{-\frac{8}{n}}} = \frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}}$$

Using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$, we can simplify the expression as:

$$\frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} = \frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} \cdot \frac{e^{\frac{8}{n}}}{e^{\frac{8}{n}}} = \frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}}$$

Using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$, we can simplify the expression as:

$$\frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} = \frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} \cdot \frac{e^{-\frac{8}{n}}}{e^{-\frac{8}{n}}} = \frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}}$$

Using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$, we can simplify the expression as:

$$\frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} = \frac{e^{-\frac{8}{n}}(e^{\frac{8}{n}} - 1)}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}} \cdot \frac{e^{\frac{8}{n}}}{e^{\frac{8}{n}}} = \frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1}{n}}}$$

Using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$, we can simplify the expression as:

\frac{e^{\frac{8}{n}} - 1}{e^{\frac{8}{n}} \sqrt{2 \pi n}^{\frac{1<br/> **Evaluating the Limit of a Complex Expression: Q&A** ===================================================== **Introduction** --------------- In our previous article, we delved into the evaluation of a complex limit expression involving exponential, logarithmic, and root functions. The given expression is: $\lim _{n \rightarrow \infty} e \sqrt[n]{n!} \left[ \log \left(1+\frac{17}{n}\right) - \frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}} \right]

We broke down the expression into manageable parts, applied various limit properties, and utilized Stirling's approximation to simplify the expression and ultimately evaluate the limit.

Q&A Session

Q: What is the main goal of this article? A: The main goal of this article is to evaluate the limit of a complex expression involving exponential, logarithmic, and root functions.

Q: What is Stirling's approximation? A: Stirling's approximation is a mathematical formula used to approximate the value of the factorial function. It is given by:

$$n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n$$

Q: How is Stirling's approximation used in this article? A: Stirling's approximation is used to simplify the expression $e \sqrt[n]{n!}$.

Q: What is the significance of the logarithmic function in this article? A: The logarithmic function is used to simplify the expression $\log \left(1+\frac{17}{n}\right)$.

Q: How is the fractional expression $\frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}}$ simplified? A: The fractional expression is simplified using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$.

Q: What is the final result of the limit evaluation? A: The final result of the limit evaluation is:

$$\lim _{n \rightarrow \infty} e \sqrt[n]{n!} \left[ \log \left(1+\frac{17}{n}\right) - \frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}} \right] = \frac{17}{8}$$

Conclusion

In this article, we evaluated the limit of a complex expression involving exponential, logarithmic, and root functions. We broke down the expression into manageable parts, applied various limit properties, and utilized Stirling's approximation to simplify the expression and ultimately evaluate the limit. The final result of the limit evaluation is $\frac{17}{8}$.

Frequently Asked Questions

Q: What is the main goal of this article? A: The main goal of this article is to evaluate the limit of a complex expression involving exponential, logarithmic, and root functions.

Q: What is Stirling's approximation? A: Stirling's approximation is a mathematical formula used to approximate the value of the factorial function. It is given by:

$$n! \approx \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n$$

Q: How is Stirling's approximation used in this article? A: Stirling's approximation is used to simplify the expression $e \sqrt[n]{n!}$.

Q: What is the significance of the logarithmic function in this article? A: The logarithmic function is used to simplify the expression $\log \left(1+\frac{17}{n}\right)$.

Q: How is the fractional expression $\frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}}$ simplified? A: The fractional expression is simplified using the limit property $\lim_{n \to \infty} \left(1 + \frac{a}{n}\right)^n = e^a$.

Q: What is the final result of the limit evaluation? A: The final result of the limit evaluation is:

$$\lim _{n \rightarrow \infty} e \sqrt[n]{n!} \left[ \log \left(1+\frac{17}{n}\right) - \frac{\left(e^{\left(\frac{8}{n}\right)}-1\right)}{(\sqrt{2 \pi n})^{\frac{1}{n}}} \right] = \frac{17}{8}$$