sidef-scripts/Math/limits_for_pi.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 09 January 2017
# Edit: 09 April 2022
# https://github.com/trizen

# Two limits for Pi: an upper and a lower limit.

# The following two inequalities hold:
#
#   lower(n) < Pi < upper(n)
#
# for n < Infinity.

# See also:
#   https://oeis.org/A280100
#   https://oeis.org/A280845
#   https://oeis.org/A134374

#
## 16^n * n * (n!)^4 / ((2*n + 1)!)^2
## (4^n * n * (n!)^2) / ((2*n + 1)!!)^2
#
func lower(n) {
    4 * (16**n * n * n!**4) / (2*n + 1)!**2
    #(4**(n+1) * n * (n!)**2) / ((2*n + 1)!!)**2
}

#
## 16^n * (n+1)! * (n!)^3 / ((2*n + 1)!)^2
## (4^n * (n+1) * (n!)^2) / ((2*n + 1)!!)^2
#
func upper(n) {
    4 * (16**n * (n+1)! * n!**3) / (2*n + 1)!**2
    #(4**n * (n+1) * (n!)**2) / ((2*n + 1)!!)**2
}

#
## 16^n * (n!)^4 / (n * ((2*n)!)^2)
## (2^n * n!)^4 / (n *  ((2*n)!)^2)
## 4^n * (n!)^2 / (n * ((2*n - 1)!!)^2)
#
func upper2(n) {
    #16**n * n!**4 / (n * (2*n)!**2)
    #4**n * (n!)**2 / (n * ((2*n - 1)!!)**2)
    (2**n * n!)**4 / (n * (2*n)!**2)
}

define LIMIT = 1e3

say "Lower bounds:"
say lower(LIMIT)

say "\nUpper bounds:"
say upper(LIMIT)
say upper2(LIMIT)

__END__
Lower bounds:
3.13923812696691110493340150080310032881057733291

Upper bounds:
3.14237736509387801603833490230390342913938791024
3.14237814990340975776611113565427862991447011289