Why do I see i and k as the indices of summation?

Question

I'm working on linear algebra and just wanted to clear up an uncertainty regarding whether there is a difference in the use of i and k as the dummy variables for the index of summation?

$\sum\limits_{i=1}^\infty {i^2} = \sum\limits_{k=1}^{\infty} {k^2}$ ?

I got confused at first since I was working with vectors [i, j] with a summation indicating k=1 , although k was indicating the z dimension (i and j indicating x and y respectively).

Just to clarify, the choice of i and k as dummy variables is completely arbitrary - right?

I found my way to this page at Wolfram MathWorld and it actually switches from i to k in the course of a short piece of text, is this normal and nothing to concern me or should I take note of differences like this?

Answer 1

Summation indices are dummy variables that are completely arbitrary.

If $i$, $j$, and $k$ are already being used for vector notation, it would be good to use a different index for summation. The letter $m$ would be one sensible choice, if you are writing things like $\sum_{m = 0}^n$, and $l$ is another possibility (just because it is close in the alphabet to $i,j,k,m,$ and $n$). Of course you are free to use any variable that hasn't already been given a meaning, but it is good to use letters that will have the psychological connotation of being an index (so letters like $x$, $y$, and $z$ are fairly uncommon as summation indices).

Answer 2

Yes, it's completely arbitrary. Although it would be frowned upon, there is nothing inherently wrong using something like $\dagger$ or $わ$ or a drawing of an acorn as dummy variables either. Although, you should try not to switch too often during the course of a text. In the case of the WMW text you link, the $k$ symbolizes the same thing (the order of the forward difference), although it changes from being the bound of one sum to being the summation variable in the next.

There's no a priori difference between the notations $$ \sum_{i=1}^\infty {i^2}\\ \sum_{k=1}^\infty {k^2}\\ \sum_{\dagger=1}^\infty {\dagger^2}\\ \sum_{わ=1}^\infty {わ^2} $$ However, if you've used one of them before, then using them again would be seen as wrong unless, as in the WMW example, it still symbolizes the same quantity.

Answer 3

The summation index is a bound or dummy variable. It doesn't matter what variable you use. In your example, the sum up to infinity is problematic because it doesn't converge, but the use of $i$ or $k$ doesn't change anything.