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?
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.
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.
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.