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Chapter 11: Q11.5P (page 554)

Use Stirling鈥檚 formula to evaluate $\underset{n 鈫� 鈭�}{l i m} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .

Short Answer

Expert verified

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

Step by step solution

01

Given Information

The function is $\lim_{n 鈫� 鈭�} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .

02

Definition of the sterling’s formula.

Sterling鈥檚 formula is used to simplify formulas involving factorial.

$n! 鈻� n^{n} e^{- n} \sqrt{2 蟺苍}$ .

03

Find the value of the function.

The function is $\lim_{n 鈫� 鈭�} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!}$ .

The sterling鈥檚 formula is $n! 鈻� n^{n} e^{- n} \sqrt{2 蟺苍}$ .

$\begin{array}{rcl} (n + \frac{3}{2})! & 鈮� & {(n + \frac{3}{2})}^{n + \frac{蟺}{2}} e^{- (n + \frac{3}{2})} \sqrt{2 蟺 (n + \frac{3}{2})} \\ (n!) & 鈮� & (n^{n} e^{- n} \sqrt{2 蟺苍}) \end{array}$

Substitute above valuesin the equation mentioned below.

$\lim_{n 鈫� 鈭�} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!} = \frac{{(n + \frac{3}{2})}^{n + \frac{蟺}{2}} e^{- (n + \frac{3}{2})} \sqrt{2 蟺 (n + \frac{3}{2})}}{n^{n} e^{- n} \sqrt{2}} \lim_{n 鈫� 鈭�} \frac{(n + \frac{3}{2})!}{\sqrt{n} (n + 1)!} = e^{\frac{- 3}{2}}$

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

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Most popular questions from this chapter

Show that for $k = 0$ , $u = F (蠒, 0) = 蠒, s n u = s i n u, c n u = c o s u, d n u = 1 .$ and for $k = 1$ $u = F (蠒, 1) = l n (s e c 蠒 + t a n 蠒), 蠒 = g d u, s n u = t a n h u, c n u = d n u = sech u$ .

Without computer or tables, but just using facts you know, sketch a quick rough graph of the $螕$ function from -2to 3. Hint:This is easy; don鈥檛 make a big job of it. From Section 3, you know the values of the data-custom-editor="chemistry" $螕$ function at the positive integers in terms of factorials. From Problem 1, you can easily find and plot the $螕$ function at $卤 1 / 2$ , $卤 3 / 2$ . (Approximateas a little less than 2.) From (4.1) and the discussion following it, you know that the $螕$ function tends to plus or minus infinity at 0 and the negative integers, and you know the intervals where it is positive or negative. After sketching your graph, make a computer plot of the 螕 function from -5to 5and compare your sketch.

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