# 1/n! * (n! + (n-1)! + (n-2)! ......1!)      # can be factored down to:

Very clever

# n!/n! + (n-1)!/n! + (n-2)!/n! .....1!/n!    # write out the factorial e.g. 5 x 4 x 3 ... and you'll see it better

#  s(n) =  1   +    1/n    +  1/n(n-1) ...1/n!       # 1/n plays the largest role. In fact, after n=15, 
                                              # the 1st 6 decimals aren't even affected - trial and error
# s(n-1) = 1 + 1/(n - 1) + ... + 1/(n-1)!

# A recursive formula indicates s(n-1)/n + 1 = 1 + 1/n + 1/(n-1)n + ... + 1/n! =s(n)

to truncate to 6 decimal places

can one explain how the main formula (1 / n!) * (1! + 2! + 3! + ... + n!) been simplified to the above solution (in a simple for loop). Also please recommend me where I can learn the tricks to simply maths to code in efficient way.

Maybe because it's truncating, not rounding?

Is there any reason not to use round here? Does this method have more efficiency here?