Recall that if n is a positive integer, then n! is the product of all the integers from 1 to n, inclusive. Show that. (2¹⁰)! > 2 ^{2^13} .

Via Stirling's approximation, and conversion to common log, log((2^10)! ~= 2639.73388138571.

log(2^(2^13)) = (2^13) * log(2) ~= 2466.037724479334.

Therefore (2^10)! > 2^(2^13).

(2^10)! ~=  5.418528796 * 10^2639
2^(2^13) ~= 1.0907481356 * 10^2466

 

Posted by Charlie on 2013-07-11 17:38:47