Summation | CAS CMS

Summation

The addition of a sequence of numbers can be represented with the summation symbol (Σ).

Consider an addition sequence, S_n. Suppose the numbers to be added are u₁, u₂, u₃, ..., u_n. We let

$\begin{align*} S_1 &= u_1\\ S_2 &= u_1 + u_2\\ S_3 &= u_1 + u_2 + u_3\\ &\ldots\\ S_n &= u_1 + u_2 + \ldots + u_n = \sum\limits_{k=1}^n u_k \end{align*}$

The sigma (Σ) symbol represents a summation of n components. As n increases without bound...

$u_1 + u_2 + \ldots + u_n + \ldots$

...we are led to consider a summation over infinite components, which is denoted by

$\sum\limits_{k=1}^{\infty} u_k$

Such an expression is called an infinite series. As an example, we can write the equation of state for a mixture of gases:

$P = \sum\limits_i P_i = \sum\limits_i n_ikT$ where P = total pressure
P_i = partial pressures of all i gas species
n_i = particle number per unit volume
k = Boltzmann's constant, and
T = temperature

This equation describes the total pressure P, as the summation of partial pressures P_i.