Algebraic rules involving series: vectorial concept

Remarks.

The operations of addition of sequence and scalar multiplication of a sequence are known as the two fundamental operations of objects called vectors, which we shall study in detail later. We are reminded of the “Big Idea of Vectors”.

It is good to put in one place all the standard algebraic rules for handling series in sigma notation:

1. \[\sum_{r=m}^{n} a = \underbrace{a + a + \dots + a}*{n \text{ terms}} = (n - m + 1)a, \text{ where}\hspace{0.5em} m \leq n.\]
2. \[\sum*{r=m}^{n} u_r = \sum_{r=1}^{n} u_r - \sum_{r=1}^{m-1} u_r, \text{ where}\hspace{0.5em} m \leq n.\]
3. \[\sum_{r=1}^{n} (u_r + v_r) = \sum_{r=1}^{n} u_r + \sum_{r=1}^{n} v_r.\]
4. \[\sum_{r=1}^{n} \alpha u_r = \alpha \sum_{r=1}^{n} u_r.\]

The above four rules can be seen as “decomposition” rules, which can be used to break down a complicated looking term into its constituents.