November 03, 2013

Horse 1565 - Infinite Zaccheus

In church this morning, before communion we looked at a passage in Luke and Zacchaeus the Tax Collector. There are all sorts of lessons that can be learnt here such as who is redeemable, the sort of attitudes that happen once we meet Christ and Christ's willingness to meet people irrespective of their place or reputation in life.
This passage can be found in Luke 19 - Link:

Really the thing that most people remember about Zacchaeus isn't the efforts he took to meet Christ or his generosity but the fact that he was short and could not see over the crowd.
Invariably there are jokes which can be made about who the shortest person in the bible was. Was it Zacchaeus or Nehemiah (knee-high-miah) or one of Job's mates, Bildad the Shuhite?
What would happen though if we took Zacchaeus, Nehemiah,  Bildad the Shuhite and placed them on top of each other?

The total height of them would be 1 + 1/3 + 1/9 + 1/27... + 1/3^infinite.
Surely if the fractions on the end are 1/3^(n+1) where n is the nth term, then this series must have a limit.

If we assume that ratio r is amount required to go from term n to term n+1 then, r = n/n-1 I think that works for any two consecutive terms. Take the first two:
r = (1/3)/1 = 1/3

or maybe the 9th and 10th:
r = (1/19683) / (1/59409) = 1/3

the sum of the first terms is going to be defined by:
S = 1 - r^(n+1) / 1 - r

S = 1 - (1+ 1/59409) / 1 - 1/3
S = 1.499974751301808

if n is infinite then r^(n+1) will tend towards zero.

S = 1 - 0 / 1 - 1/3
S = 1 / 1/3
S = 3/2

So placing Zacchaeus, Nehemiah,  Bildad the Shuhite and an infinite number of increasing short people in the ratio of 1/3 on top of each other means that the top of the very top most person can not be any more than 1.5 times the height of Zacchaeus.
If Zacchaeus was a head shorter than an average person today, they he'd be about 5'4" tall and as such, this tower of people would only be about 8ft tall. Mind you, even if there was such an 8ft tower of an infinitive number of increasingly smaller people, Zacchaeus would have still been at the bottom of it all and would have still been to short to see anything.

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