DATA 605 Week06 Discussion

Combinatorics and Conditional Probability (Section 3.1) - Questions 9

Question (9):

Solution:

print (exp(1))

## [1] 2.718282

print (pi)

## [1] 3.141593

for (n in 1:9) {

  A <- sqrt(2*pi*n)
  B <- (n/exp(1))^n
  C <- (exp(1))^(1/(12*n+1))
  D <- (exp(1))^(1/(12*n))
  
  lTerm <- A * B * C
  mTerm <- factorial(n)
  rTerm <- A * B * D

  comp <- (lTerm < mTerm) & (mTerm < rTerm)
  print (c(lTerm, mTerm, rTerm))
  print (comp)
}

## [1] 0.9958702 1.0000000 1.0022744
## [1] TRUE
## [1] 1.997320 2.000000 2.000652
## [1] TRUE
## [1] 5.996096 6.000000 6.000599
## [1] TRUE
## [1] 23.99082 24.00000 24.00102
## [1] TRUE
## [1] 119.9699 120.0000 120.0026
## [1] TRUE
## [1] 719.8722 720.0000 720.0092
## [1] TRUE
## [1] 5039.335 5040.000 5040.041
## [1] TRUE
## [1] 40315.89 40320.00 40320.22
## [1] TRUE
## [1] 362850.6 362880.0 362881.4
## [1] TRUE

We see that the inequality is valid for \(n\) = \(1\) to \(9\).