**Lizzy and Cheryl, 7th graders,
find the sum of an infinite series- 2 ways!**

"Using squares (see Mr. Cohen's chapter 1) we found what these infinite series went to:

Let's say you have the problem 2/5 + (2/5)^{2
}+ (2/5)^{3 }+ ..., what would you do? Normally you would do all
the work and spend lots of time on the problem, but what if there was a
shortcut? You could go on forever with this problem by just increasing the
number of terms. Well, if you want to do it in a flash, then use the simple
rule A/(B-A).. (the girls spent a lot of
time developing this rule), where
A is the numerator, B is the denominator of the
original fraction. So 2/5 + (2/5)^{2 }+ (2/5)^{3 }+ ... would go
to 2/(5-2) = 2/3.

**Another way to do this was done by one of Mr.
Cohen's students, Jeff. Jeff's idea was to change the original fraction into one
that has 1 for the numerator. When you use Jeff's way 2/5 becomes 1/2.5 .
Then you can use the other technique above. When the problem has 1 for the
numerator, you can use the rule that it goes to 1/(n-1), in this case, 1/(2.5-1)
= 1/1.5 = 2/3 ".**

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