Mathematical Notation to Represent Cumulative Sum

Question

If I have a vector $x$

$$ x = [x_1, x_2, x_3, \ldots, x_N]$$

Is there any proper mathematical notation that best represents the cumulative sum?

$$ y = [x_1, x_1+x_2, x_1+x_2+x_3, \ldots, x_1+x_2+x_3+\ldots+x_N]$$

Answer 1

Consider an infinite-dimensional vector that is trivial after some N. Then y is considered S_n the sequence of partial sums.

Answer 2

Each component will be of the form

$y_k=\sum_{i=1}^k x_i$.

Answer 3

If you use $a[i]$ to mean the $i$th element of a vector $a$, then you could define $y$ as:

$$y[i] = \sum_{k=1}^i x[k], \text{ for } i=1, 2, \dots, N$$

Answer 4

$$y = \begin{bmatrix} y_1 \\ \vdots \\ y_N\end{bmatrix}$$

where for $i \in \{ 1, \ldots, N \}$,$y_i = \sum_{j=1}^i x_j$.

Answer 5

One possible way to define it is: if $\mathbf v:=\langle v_1,v_2,\ldots,v_n\rangle$ is a vector (in an $n$-dimensional space) where the canonical projections are defined by

$$\pi_k(\mathbf v):=v_k$$

then you can define de vector $\mathbf w$ as

$$\mathbf w:=\left\langle \pi_1(\mathbf v),\pi_1(\mathbf v)+\pi_2(\mathbf v),\ldots,\sum_{k=1}^n\pi_k(\mathbf v)\right\rangle$$

Answer 6

$y=Ax$ where \begin{equation} A= \begin{bmatrix} 1&0&0&...&0&0 \\ 1&1&0&...&0&0 \\ 1&1&1&..&0&0 \\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots \\ 1&1&1&...&1&0 \\ 1&1&1&...&1&1 \end{bmatrix} \end{equation}