By Soo Jin Lee and Jaehong shin
Last day we stopped at the sum of infinite series of (1/3).
However we could conjecture the answer will be (1/2) by previous examples.
(If you cannot remember, click here to look at the day1 and day 2 activity)
Today we will going to do this activity by using a piece of square grid paper.
Outline a 27 * 27 square. Divide the square into three rows, each 9 * 27. Color in the bottm row. Now divide the middle row into three equal colomns and color in the right column. Now color in the bottom third of the middle column. Then divide the middle row above it into three equal columns and color in the right column. Continue this pattern.
another way to approach this problem is by using the powerful tool EXCEL!!
You will obtain this following picture by excel.
----this color is 1/3 of the whole square......this color is (1/3)^2 of the whole square.
.....this color id (1/3)^3 of the whole square.
By doing the similar process with (1/3)^4, (1/3)^5 ......., we will come up with the below picture.(this doesn't mean that it is the end of the sum..we can keep color if we have space.)
Do you see that every colored section has a corresponding uncolored one? That means that the sum of all the colored sections is one half.
But what about the small square in the middle of the figure. Think of that square as being a small copy of the large square. If you magnified that square, it would be colored in the same pattern as the larger one. If you repeated the coloring process infinitely many times, exactly one half that square would be colored.
click here to explore by yourself with excel.
It would now be a good conjecture that given
c is a natural number,
1/c + (1/c)2 + (1/c)3 + . . . + (1/c)n + . . . = 1/(c-1)
Next class, we will rigorously prove our conjecture.