Why does nan ind pop up when expanding the Fourier series?

Question

C++ / C#

I began to learn the pluses, such a task, to expand the function using the Fourier series, the accuracy is 0.0001.
The value of the function is found, but the row does not work

#include <iostream>
#include <cmath>

using namespace std;

void main() {
  setlocale(LC_ALL, "Russian");
  cout << "Введите значение аргумента: ";
  double x;
  cin >> x;
  const double pi = atan(1) * 4;
  double function = (pi - x) / 2;
  cout << "Значение функции: " << function << endl;
  double sum = 0;
  double i = 0;
  do {
    sum += sin(i * x) / i;
    i++;
  } while (abs(function - sum) > 0.0001);
  cout << "Значение суммы ряда по функции: " << sum;
}

Answer 1

Something does not add up in the task.
Let's take x = 0
F(0) = (π - 0) / 2 = π/2
Σ _{i = [1, ∞)} (sin(i * 0) / i) = 0