I'm looking at the summation term in the denominator. It says to take the difference of vectors x and y, and then the absolute value ... which I believe you mean element-wise(?), and then sum element-wise(?). If so, then we're good so far.
Then I gather you'd like a finite or infinite series expansion of the rational expression. Sure, that's doable and there are many types of expansions to consider. Did you have any particular one in mind?
Thanks a lot for your quick and informative reply! At first, I am happy to see it's doable. Yes, It's element-wise (that is, Manhattan distance between x and y). Honestly, I am curious to see different types of expansion and select the one more applicable to me. Anyway, to be more specific, I consider infinite series is my first choice. Thanks again for being so helpful.
The Manhattan distance is a built-in function with Mathematica: https://reference.wolfram.com/language/ref/ManhattanDistance.html?q=ManhattanDistance
There is a guide to Series in Mathematica here: https://reference.wolfram.com/language/tutorial/SeriesLimitsAndResidues.html
As far as distance functions go, I rate Manhattan (the L1 distance) as "passable" and Chessboard (the L-infinity) the worst. If you are using Manhattan for a school project then so be it.
Thank you so much for your help. I will try my best to use your given link (although I have never used wolfram before). I am using the WOLFRAM alpha and trying to do this.