\(\newcommand{L}[1]{\| #1 \|}\newcommand{VL}[1]{\L{ \vec{#1} }}\newcommand{R}[1]{\operatorname{Re}\,(#1)}\newcommand{I}[1]{\operatorname{Im}\, (#1)}\)

Some algebra with summation

We use the symbol \(\Sigma\) for summation.

Say we have a series of four values \(x_1, x_2, x_3, x_4\).

We can write the sum \(x_1 + x_2 + x_3 + x_4\) as:

\[\Sigma_{i=1}^{4} x_i\]

You can read this summation as “the sum of values \(x\) subscript \(i\) from \(i=1\) through \(i=4\)“.


\[\Sigma_{i=1}^{4} x_i = x_1 + x_2 + x_3 + x_4\]

When the indices of the summation are obvious, they may quietly disappear. For example, it may be obvious that we are summing over all \(i = 1, 2, 3, 4\), in which case we could write the \(\Sigma\) with or without the indices:

\[\Sigma_{i=1}^{4} x_i = \Sigma x_i\]

Algebra of sums

Addition inside sum

Say we have two series of numbers \(x_1, x_2 \cdots x_n\) and \(y_1, y_2 \cdots y_n\).

\[\begin{split}\Sigma_{i=1}^n (x_i + y_i) = \\ (x_1 + y_1) + (x_2 + y_2) + \cdots (x_n + y_n) = \\ (x_1 + x_2 + \cdots x_n) + (y_1 + y_2 + \cdots y_n) = \\ \Sigma_{i=1}^n x_i + \Sigma_{i=1}^n y_i\end{split}\]

Multiplying by constant inside sum

\[\begin{split}\Sigma c x_i = \\ c x_1 + c x_2 + \cdots c x_n = \\ c (x_1 + x_2 + \cdots x_n) = \\ c \Sigma x_i\end{split}\]

Sum of constant value

\[\Sigma_{i=1}^n c = n c\]