Michael, a 5th grader, finds a Pattern in Infinite Series

After coloring squares for  -> 1 and -> 1/2 and 1/4+(1/4)^2+...-> 1/3, he came up with the generalization that 1/n + (1/n)^2 + (1/n)^3 + (1/n^4)… approaches the limit 1/(n-1).

Don showed Michael how to do this in Mathematica® and Derive. Michael  tried things to show how his rule works for big numbers..

The Greek symbol Sigma means sum. So this reads find the sum of 1/3 to the nth power as n goes from 1 to 2, to 3, ... to infinity and looks like this below

which approaches 1/2 as the limit.  Michael did the following:

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Don has found that many of his students try to see what happens when they push the envelope!

A week or two later, Don asked Michael to see what happens if we start with 2/5.

Michael did a number of infinite sums, starting with

2/5 , which went to 2/3 and

74/75,  which went to 74 and

7/676767677665883, which went to 7/676767677665876 and others,

and he figured out that

A/B + (A/B)^2 + (A/B)^3 + … = A/(B-A)

He also found, by himself, that the infinite sum, starting with 89/6 gets bigger and bigger, it diverges. and starting with 3/3, the series would also diverge, does not approach a number. So they talked about how A and B were related.

 Michael said for the infinite geometric series to converge, 

A must be < B .

Fine work Michael!!!