Very small number turns negative in Fortran - fortran

Question

I'm doing a program in Fortran90 which do a sum from i=1 to i=n where nis given. The sum is sum_{i=1}^{i=n}1/(i*(i+1)*(i+2)). This sum converges to 0.25. This is the code:
PROGRAM main
INTEGER n(4)
DOUBLE PRECISION s(4)
INTEGER i
OPEN(11,FILE='input')
OPEN(12,FILE='output')
DO i=1,4
READ(11,*) n(i)
END DO
PRINT*,n
CALL suma(n,s)
PRINT*, s
END
SUBROUTINE suma(n,s)
INTEGER n(4),j,k
DOUBLE PRECISION s(4),add
s=0
DO k=1,4
DO j=1,n(k)
add=1./(j*(j+1)*(j+2))
s(k)=s(k)+add
END DO
END DO
END SUBROUTINE
input
178
1586
18232
142705
The output file is now empty, I need to code it. I'm just printing the results, which are:
0.249984481688 0.249999400246 0.248687836759 0.247565846142
The problem comes with the variable add. When j is bigger, add turns negative, and the sum doesn't converge well. How can I fix it?

Answer 1

The problem is an integer overflow. 142705142706142707 is a number that is too large for a 4-byte integer.
What happens then is that the number overflows and loops back to negative numbers.
As #albert said in his comment, one solution is to convert it to double precision every step of the way: ((1.d0/j) / (j+1)) / (j+2). That way, it is calculating with floating point values.
Another option would be to use 8-byte integers:
integer, parameter :: int64 = selected_int_kind(17)
integer(kind=int64) :: j
You should be very careful with your calculations, though. Finer is not always better. I recommend that you look at how floating point arithmetic is performed by a computer, and what issues this can create. See for example here on wikipedia.

Answer 2

This is likely a better way to achieve what you want. I did remove the IO. The output from the program is
% gfortran -o z a.f90 && ./z
178 0.249984481688392
1586 0.249999801599584
18232 0.249999998496064
142705 0.249999999975453
program main
implicit none ! Never write a program without this statement
integer, parameter :: knd = kind(1.d0) ! double precision kind
integer n(4)
real(knd) s(4)
integer i
n = [178, 1586, 18232, 142705]
call suma(n, s)
do i = 1, 4
print '(I6,F18.15)', n(i), s(i)
end do
contains
!
! Recursively, sum a(j+1) = j * a(j) / (j + 1)
!
subroutine suma(n, s)
integer, intent(in) :: n(4)
real(knd), intent(out) :: s(4)
real(knd) aj
integer j, k
s = 0
do k = 1, 4
aj = 1 / real(1 * 2 * 3, knd) ! a(1)
do j = 1, n(k)
s(k) = s(k) + aj
aj = j * aj / (j + 3)
end do
end do
end subroutine
end program main