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Chapter 5: Q27P (page 239)

(a) Find the percent error in Stirling’s approximation for z = 10 ?

(b)What is the smallest integer z such that the error is less than 1%?

Short Answer

Expert verified

(a) Error of Stirling's approximation for z =10 is 13.75%.

(b) Error of Stirling's approximation is less than 1% for z = 90.

Step by step solution

01

Define the Stirling’s approximation

  • Stirling's approximation (or Stirling's formula) is a factorial approximation in mathematics.
  • It is a good approximation, yielding accurate results even for low displaystyle nn values. It takes its name from James Stirling.
02

Calculating the percent error

(a)

For z = 10 we have

In(10!)=15.104410In(10)-10=13.0259%Error=15.1044-13.025915.104413.76%

Therefore the percent error in Stirling’s approximation is 13.76% .

03

Calculating the smallest integer

(b)

% Error is calculated by:

In(z!)zIn(z)+zIn(2!).100z%Error215.671000.89900.996851.06891.009

Therefore we see that for z = 90 and more error is less than 1%.

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