Array.Sum() results in an overflow

You will have to cast it to long so you don't overflow

var result = arr.Select(x => (long)x).Sum();

int (C# Reference)

Range = -2,147,483,648 to 2,147,483,647

Some background, this is the source code for Sum

public static int Sum(this IEnumerable<int> source) 
{
      if (source == null) throw Error.ArgumentNull("source");
      int sum = 0;
      checked 
      {
          foreach (int v in source) 
             sum += v;
      }
      return sum;
}

Meaning, whether you like it or not, someone is using a for loop, additionaly the usage of checked is why it throws :)

checked (C# Reference)

The checked keyword is used to explicitly enable overflow checking for integral-type arithmetic operations and conversions.

You can find sum as shown below

using System;

public class Program
{
    public static void Main()
    {
        Console.WriteLine("Hello World");
        int[] arr = {999999999, 999999999, 999999999,999999999, 999999999};
         long sum = 0;
         for (int i = 0; i < arr.Length; i += 1)
         {
           sum += arr[i];
         }
        Console.WriteLine(sum);
    }
}

Datatype and Range Reference

When you call sum on array of integer, output type is an integer, and sum of entered numbers are more than integer maxvalue. as @TheGeneral said sum method check overflow and throw exception.

may be you expect the sum method return result as long, but its return type is an integer.

int a = 256741038;
int b = 623958417;
int c = 467905213;
int d = 714532089;
int e = 938071625;

long sumInt = a + b + c + d + e;
//-1293758914 //overflow happened withoud exception
long sumLong = (long)a + b + c + d + e;//clr now the result may be more than int.maxvalue
//3001208382

A shorter way is as shown below

using System;
namespace Test
{
    public partial class TestPage : System.Web.UI.Page
    {
        public int[] p = { 999999999, 999999999, 999999999, 999999999, 999999999 };
        protected void Page_Load(object sender, EventArgs e)
        {
            long s = Sum;
        }
        public long Sum
        {
            get { long s = 0; Array.ForEach(p, delegate (int i) { s += i; }); return s; }
        }        
    }
}

Hope this will help you.