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

Consider an addition sequence, Sn. Suppose the numbers to be added are u1, u2, u3, …, un. We let

\begin{align*} <br />
S<em>1 &= u</em>1\\<br />
S<em>2 &= u</em>1 + u_2\\<br />
S<em>3 &= u</em>1 + u_2 + u_3\\<br />
&\ldots\\<br />
S<em>n &= u</em>1 + u<em>2 + \ldots + u</em>n = \sum\limits_{k=1}^n u_k

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
Pi = partial pressures of all i gas species
ni = 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 Pi.

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