Algebraic rules involving series: constant series

We now develop the algebraic rules to handle series using sigma notation.

First, we study the sum of a constant sequence \(u_r = a\) (\(r = 1, 2, 3, \dots\)), where \(a\) is a fixed constant that does not change with \(r\). As we write out the first \(n\) terms of the constant sequence we have:

\[\underset{=u_1}{a}, \underset{=u_2}{a}, \dots, \underset{=u_n}{a}\]

If we add them up, we have

\[a + a + \dots + a = na.\]

Thus, we gather that \(\sum_{r=1}^{n} a = na\).

In general, if the running index ranges from \(m\) to \(n\) (where \(m \le n\)) then

\[\sum_{r=m}^{n} a = \underbrace{a + a + \dots + a}_{n-m+1 \text{ terms}} = (n - m + 1)a.\]