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Some algebra with summation
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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:

.. math::

    \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:

.. math::

    \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:

.. math::

    \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 $y_1, y_2 \cdots
y_n$.

.. math::

    \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

.. _multiply-inside-sum:

Multiplying by constant inside sum
==================================

.. math::

    \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

Sum of constant value
=====================

.. math::

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

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More reading
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* `A basic tutorial on summation with tests
  <http://www.psychstat.missouristate.edu/IntroBook3/sbk10.htm>`_;
* A list of `summation identities`_.


.. include:: links_names.inc