Why is the result false when I run this? Where is the error?

I made a few small changes and the result seems correct to me. The calculation loop look likes this :

double s = 0;
for( int i = 1; i <= length; i++ )
{
    s += ( 1.0 / ( i * ( i + 1 ) ) );
}

The changes were :
1. Use a double for the calculation result.
2. Cast to a double for the denominator.
3. Use a double for the numerator.

What you do did results in dividing two integers and you will not get the correct result doing that.

ETA: item 2 is unnecessary. Using a double for numerator is enough to give a double for a result.

I would write it this way (see the comments)

//Tinh tong Sn = 1/1*2 + 1/2*3 + 1/3*4 + ... + 1/n*(n+1)
  
#include<stdio.h>
int main()
{
  int n,i;
  double s = 0.0; // unless you have a very good reason to use 'float', use 'double' instead
  do
  {
    printf("n=");
    scanf("%d", &n); // here you have to pass the address of 'n'
  }
  while(n<=0);
  for(i = 1; i <= n; i++)
  {
    s += (1/(i*(i+1.0))); // using 1.0 forces the floating point evaluation of the expression
  }

  printf("\nS(%d) = %g\n", n, s);

  return 0;
}

Just to add to what the others have said, you should realize that you are going to get problems like this a lot: they are called "bugs" and detecting, analysing, and fixing bugs is a major part of the development process - compiling does not mean your code is right! :laugh:

Think of the development process as writing an email: compiling successfully means that you wrote the email in the right language - English, rather than German for example - not that the email contained the message you wanted to send.

So now you enter the second stage of development (in reality it's the fourth or fifth, but you'll come to the earlier stages later): Testing and Debugging.

Start by looking at what it does do, and how that differs from what you wanted. This is important, because it give you information as to why it's doing it. For example, if a program is intended to let the user enter a number and it doubles it and prints the answer, then if the input / output was like this:

Input   Expected output    Actual output
  1            2                 1
  2            4                 4
  3            6                 9
  4            8                16

Then it's fairly obvious that the problem is with the bit which doubles it - it's not adding itself to itself, or multiplying it by 2, it's multiplying it by itself and returning the square of the input.
So with that, you can look at the code and it's obvious that it's somewhere here:

int Double(int value)
   {
   return value * value;
   }

Once you have an idea what might be going wrong, start using the debugger to find out why. Put a breakpoint on the first line of the method, and run your app. When it reaches the breakpoint, the debugger will stop, and hand control over to you. You can now run your code line-by-line (called "single stepping") and look at (or even change) variable contents as necessary (heck, you can even change the code and try again if you need to).
Think about what each line in the code should do before you execute it, and compare that to what it actually did when you use the "Step over" button to execute each line in turn. Did it do what you expect? If so, move on to the next line.
If not, why not? How does it differ?
Hopefully, that should help you locate which part of that code has a problem, and what the problem is.
This is a skill, and it's one which is well worth developing as it helps you in the real world as well as in development. And like all skills, it only improves by use!