**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

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!!!