What is it called when you replace a sum function?

Question

Let's say I have the following sum:

$$\sum^{x}_{n=1}{n}$$

This can obviously be replaced with Gauss's formula, which doesn't use a sum function or any inherent 'looping':

$$\frac{x(x+1)}{2}$$

Similarly, we can do this with summing of squares:

$$\sum^{x}_{n=1}{n^2}=\frac{x(x+1)(2x+1)}{6}$$

And many more (although not all) series. Is there a name for this process of getting rid of the sum for a non-loopy formula? I was thinking 'linearisation', but that seemed too ambiguous, and misleading as it often involves squaring numbers (i.e making them less linear).

Answer 1

We generally call it the Closed Form of the expression . So $\dfrac{n(n+1)}{2}$ is the closed form of $\sum_{k=1}^n k$ .