Please do all parts, clearly label and separate each part. please write clearly

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Please do all parts, clearly label and separate each part. please write clearly or type.

that TION PROJECT la +b) Cantor's Disappearing Table is defined The following procedure shows how to make a table disappear by removing only half of the table! (a) Original table has a length of L let Prove (b) Remove of the table centered at the midpoint. Each . for remaining piece has a length that is less than L (c) Remove of the table by taking sections of length T from the centers of each of the two remaining pieces. Now you have removed + ? of the table. Each remaining piece has a length that is less than L d) Remove tó of the table by taking sections of length l from the centers of each of the four remaining pieces. Now you have removed1of the table. Each remaining piece has a length that is less than 81. se, in Will continuing this process cause the table to disappear, even though you have removed only half of the table? Why? FOR FURTHER INFORMATION Read the article "Cantor's Disappearing Table" by Larry E. Knop in The College Mathematics Journal. To view this article, go to MathArticles.com.

Explanation / Answer

The exact location from where a portion of the table is being removed, is illustrated in the picture.
In the first step
Table removed = L/4
Total table removed = L/4
Table remaining = 3L/4

In the second step
Table removed = L/8
Total table removed = L/4+L/8 = 3L/8  
Table remaining = 5L/8

In the third step
Table removed = L/16
Total table removed = L/4+L/8+L/16 = 7L/16  
Table remaining = 9L/16

This process will be continued. We notice that after each successive step, the portion of the remaining table is decreasing as some part of the table is being removed. However, it is noteworhty that in each successive step, the poriton of the table being removed is also decreasing by a factor of '1/2'

Thus, total length rmeoved if the process goes on forever = L/4+ L/8 + L/16+.....
Now we will have to check if this sum eventually becomes =L and if it does then the table will disappear.
L/2+L/4+L/8+L/16 .... forms an infintie Geometric Progression with first term (a) = L/4 and common ratio (r) =1/2
Thus, sum of an infinite GP = a/(1-r) = L/4 / (1-1/2) = L/4/(1/2) = L/4 * 2 = 2L/4 = L/2;
Thus, if the process continues forever, then the total length of the table removed = L/2;

Hence, we can say that the table will not disappear.

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