h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

As regards this Zeno Paradox, The Guide Entry Researcher proposes to explain it by stating that what the paradox confuses is the infinite divisibility of space vs the infinite extendability thereof i.e. if infinitly divisible time is juxtaposed upon the infinitly devisible space,clarity of resolution will speak for itself- and that I take leave to doubt.In your notation it is expresible as the sleeping 8, that proxies for space, reclining upon another somniferous eight,standing for time.
With all due respect,the proper analogy for it is akin to prisoners huddled together on a double-decker bed, locked away from the redeeming rays of daylight,oblivious of the sun of clarity.

My proposition to resolve the Zeno paradox is rooted in the idea that though we have to walk the infinite amount of legs of the journey to reach the opposite door, each following leg of our journey will be twice shorter the the former one. i.e. to reach the door I need to walk all the distance to it; the acomplisment whereof ,on the other hand, requires me to reach half the distance first,than the quarter of that distance,eighth, sixthteenth part of it and so forth ad infinitum.
But at the moment when the task to approach the door seems absolutely daunting and unacomplishable , I am reminded of the fact that what I have in hand is no more than the receding geometrical progression: 1+1/2+1/4+1/8... And though the number of such additives in the progression be infinite, our math schoolbooks tell us that their sum is strictly finite and determinable.
I have no reason to disbelieve those textbooks and ,therefore, regain my shaken confidence that planes will travel through the sky, trains will conect distant places far and near, and my door will not remain shut to the knockings of friends for long.
But, however,I find it always jarring in me that finity can be engendered by infinity.For that reason it is always a miracle that we can use integral and difirential calculus so safely and that it stands us in such a good stead.

Yes, the sum is finite, but you need to go through an infinite number of iterations to reach it. (1/2 + 1/4 + 1/8 +...) *does* add up to a whole room, but that's the physical distance. The *time* it takes is the number of additives in the geometrical progression, ie infinite strides; so infinite seconds, infinite millenia, whatever.

The Entry is right, but not clear. The simple solution is to remember that your stride, at least in most cases, does not shrink as you walk. You *will* cross the room in... well, about half a dozen strides in the case of my living room, and not an infinite number! Even his fellow Greeks with their somewhat obtuse numbering system should have set poor Zeno straight...

yes this is a convergant series in common mathematical terms it is a converging P series. basic form of (1/x)^p these converge for all p>2

For p > 1, even, if I remember correctly.