Dummy variable in Sigma notation

Remarks.

1. Why do we need the sigma notation? Well, the sigma notation is a very compact and convenient way to represent an otherwise lengthy series \(u_m + u_{m+1} + u_{m+2} + \dots + u_n\) in the space-saving and elegant looking form \(\sum_{r=m}^{n} u_r\).
2. This compact sigma form also allows us to perform algebraic manipulations very easily as we shall see later.
3. Note that the variable that stands for the running index in the sigma notation does not appear in the expanded form of the series. Thus, if we change the variable of the running index to another one the final expanded form remains the same. More precisely,

\[\sum_{r=1}^{10} r^2 = 1^2 + 2^2 + 3^2 + \dots + 10^2 = \sum_{k=1}^{10} k^2,\]

no matter whether we are using the variable \(r\) or \(k\) for the running index. Because it does not matter which variable we use for the running index, such a variable is called a dummy variable.