
# 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 $$i=4$$”.

So:

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

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¶

$\Sigma_{i=1}^n c = n c$