Continuing from my previous post about what I learned from the PCMI morning math problem sets, this post will focus on various proofs of the geometric series formula. Variations on the geometric series came up many times as we were calculating expected value.
To calculate the sum of a geometric series
the following is true:
\(1+\frac {1}{b} + \frac
{1}{b^2}+ \frac {1}{b^3}... = \sum_{n=0}^{\infty}r^n = \frac{1}{1-r}\) where
\(\frac{1}{b}=r\) and \(|r|<1\).
a)
Paper splitting: Say that there are
b people and one of the people, the “dealer”, starts with b papers. The dealer
proceeds to give each of the other people a paper and keep one for herself. She
then splits her paper into b parts, and gives each of the other people one part
and keeps one part for herself. She repeats the process until she has no more
paper and therefore, her piece of paper is evenly split between the other \(
b-1\) people. Each other person gets
\(1+\frac {1}{b} + \frac {1}{b^2}+ \frac {1}{b^3}...\) pieces of paper,
which has to equal the original piece they got plus the \(\frac{1}{b-1}\)they
were given of the dealer’s paper. Then, if you don’t believe intuitively that
\(1+\frac{1}{b-1}=\frac{1}{1-\frac{1}{b}}\), here’s algebraic proof: \(1+\frac{1}{b-1}=\frac{b-1}{b-1}+
\frac{1}{b-1}=\frac{b}{b-1}=\frac{1}{\frac{b-1}{b}}=\frac{1}{\frac{b}{b}-\frac{1}{b}}=\frac{1}{1-\frac{1}{b}}\).
And here’s a beautiful visual proof
that my table group created for when b=4.
b)
Algebraic substitution:
\(S=1+r+r^2+r^3+r^4…\)
\(S=1+ r(1+r+r^2+r^3…)\)
\(S=1+rS\) (Substitute S into the
equation above)
\(S-rS=1\)
\(S(1-r)=1\)
\(S=\frac{1}{1-r}\)
c)
Algebraic elimination:
\(S=1+r+r^2+r^3+r^4…\)
\(rS=r+r^2+r^3+r^4…\) (Scale the
original equation by r)
\(S-rS=1\)
\(S(1-r)=1\)
\(S=\frac{1}{1-r}\)
d)
Recursion:
A dog’s bowl starts filled with 1 liter of water. By the end
of each day, the dog has \(\frac{1}{b}\) of the water it started the day with
and the owner adds another liter of water. If this process continues forever,
how much water will the bowl eventually always start with?
|
Days
n
|
Amount of water
W(n)
|
|
0
|
1
|
|
1
|
\(\frac{1}{b} + 1\)
|
|
2
|
\(\frac{1}{b^2} +\frac{1}{b}+1\)
|
|
3
|
\(\frac{1}{b^3} +\frac{1}{b^2} +\frac{1}{b}+1\)
|
Recursive formula: \(W(n) = \frac{1}{b}*W(n-1)+1\)
Since we know that this will lead to a steady state,
eventually…
\(W = \frac{1}{b}*W+1\)
\(W-\frac{1}{b}*W=1\)
\(W(1-\frac{1}{b})=1\)
\(W=\frac{1}{1-\frac{1}{b}}\)
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