Summation with exclusion

Question

Commonly we use $\sum$ symbol to compactly describe the sum of a collection or a series, e.g. $\sum_{i=1}^{5} = 1 + 2 + 3 + 4 + 5$. Is it possible to describe the summation with a number removed from the series, like $1+2+4+5$, is it possible to use $\sum$ to describe this summation?

Answer 1

Here are two ways: $$\sum_{\substack{i=1\\i \ne 3}}^5 i$$ $$\sum_{i \in \{1,\dots,5\}\setminus \{3\}} i$$

If the ground set $\{1,\dots,5\}$ is clear from the context, you can even get by with just $$\sum_{i \ne 3} i$$

Answer 2

Sometimes it's more convenient to write

$$x_1+\ldots+\widehat{x_j}+\ldots+x_n$$

which is another way to denote

$$\underset{i\ne j}{\underset{1\le i\le n}{\sum}}x_i.$$