Chapter 11: Q11.5P (page 554)

Use Stirling’s formula to evaluate $\underset{n \to \infty}{l i m} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .

Short Answer

Expert verified

The value of the function is $e^{\frac{- 3}{2}}$ .

Step by step solution

Given Information

The function is $\lim_{n \to \infty} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .

Definition of the sterling’s formula.

Sterling’s formula is used to simplify formulas involving factorial.

$n! □ n^{n} e^{- n} \sqrt{2 πn}$ .

Find the value of the function.

The function is $\lim_{n \to \infty} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .

The sterling’s formula is $n! □ n^{n} e^{- n} \sqrt{2 πn}$ .

$\begin{array}{rcl} (n + \frac{3}{2})! & \approx & {(n + \frac{3}{2})}^{n + \frac{π}{2}} e^{- (n + \frac{3}{2})} \sqrt{2 π (n + \frac{3}{2})} \\ (n!) & \approx & (n^{n} e^{- n} \sqrt{2 πn}) \end{array}$

Substitute above valuesin the equation mentioned below.

$\lim_{n \to \infty} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!} = \frac{{(n + \frac{3}{2})}^{n + \frac{π}{2}} e^{- (n + \frac{3}{2})} \sqrt{2 π (n + \frac{3}{2})}}{n^{n} e^{- n} \sqrt{2}} \lim_{n \to \infty} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!} = e^{\frac{- 3}{2}}$

Hence, the value of the function is $e^{\frac{- 3}{2}}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use Stirling’s formula to evaluate $\underset{n \to \infty}{l i m} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .

Short Answer

Step by step solution

Given Information

Definition of the sterling’s formula.

Find the value of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Quantum Physics

Force

Astrophysics

Waves Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Use Stirling’s formula to evaluatelimn→∞(n+32)!n(n+1)!.

Short Answer

Step by step solution

Given Information

Definition of the sterling’s formula.

Find the value of the function.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Quantum Physics

Force

Astrophysics

Waves Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Use Stirling’s formula to evaluate $\underset{n \to \infty}{l i m} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .