Mahmudul, 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.

Hi Mahmudul, 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?