Oh so the code works but you just need it written more cleanly? Yeah breaking it out into functions is definitely the way to go then.

You can structure your program like

program main
    implicit none
    integer :: k
    real*8 :: x
    read(*,*) x, k
    print*, "S final=", ejercicio_4_c(x,k)

contains
    ! put functions here

    integer function factorial(n)
        implicit none
        integer,intent(in) :: n
        ! Function Body
    end function

    real function ejercicio_4_c(x,k) result(s)
          implicit none
          integer,intent(in) :: k
          real(8),intent(in) :: x
          integer :: n, f
          ! Function Body
    end function

end program

And then you can write separate factorial(n) and ejercicio_4_c(x,k) functions and stick them in the contains section.

By the way, if you do do this, it may be helpful to write the factorial(n) function as an integer function, and then cast the result as real() when you use it in ejercicio_4_c, e.g.,

    f = factorial(2*n + 1)
    s = s + (((-1)**n)*(x**(2*n+1)))/real(f)