Summation Operator

#beginner #intermediate

You may have seen the following mathematical symbol $Σ$ before, also known as the summation operator. For those familiar with the properties of the summation or product operator you might be familiar with the general form of summation:

\begin{aligned} \sum_{i = 0}^{n} x_{i} = x_{1} + x_{2} + x_{3} + \dots + x_{n} \\ i = index of summation operator \\ n = upper limit of summation operator \end{aligned}

Translating to Code

The summation operator is effectively just a for loop where you take the sum of the elements in each iteration.

float sum = 0;
float[] x;
int n = x.Length;

for (int i=0, i<=n, i++)
{
	sum += x[i];
}

However, although this is useful, we don't get a simple case like this all the time. We need to be able to adapt this for different summations. Let's take a look at the example:

S = \sum_{i = 0}^{10} i = 0 + 1 + 2 + 3 + \dots + 10 = 55

float sum = 0;
int n = 10;

for (int i=0; i<=n; i++)
{
	sum += i;
}

We are also often looking at more complex and general cases than this where you need to take the sum of a function. For example:

S = \sum_{i = 0}^{n} f (x_{i}) = f (x_{1}) + f (x_{2}) + \dots + f (x_{n})

float sum = 0;
float[] x;
int n = x.Length;

for (int i=0; i<=n, i++)
{
	float sillyStiringFuel = x[i];
	
	sum += DispensedSillyString(sillyStiringFuel);
}

Console.WriteLine("Total Silly String Length: " + sum);

Additional Resources

For more learning on the topic: