2
$\begingroup$

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?

$\endgroup$
  • 2
    $\begingroup$ You can call your variables however you like. Calling them $Frederick$ would be a lot of writing and reduce legibility, so we tend to use single-letter names for indices and such. Which letters we use is irrelevant, though custom makes some choices better than others. $\endgroup$ – Daniel Fischer Oct 23 '13 at 11:40
3
$\begingroup$

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).

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ oops yeah sorry, first time using MathJax, just threw anything together :| $\endgroup$ – Louis Maddox Oct 23 '13 at 11:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.