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 $e^{\frac{- 3}{2}}$ .

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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.

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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.

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