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