Page 178 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

is a monotonically decreasing sequence, because

p
< 1
q

means that

p p p p
n+1 n n
= ∗ < .
q q q q

So, if we find N ∈ N such that

p
N
<,
q

we’ll have found the giant number N, because for all n> N,

n N
p p
n
|x| < < <.
q q

Now, because p< q, there is some m ∈ N such that

q = p + m.

So, we can re-arrange

N
N
q =(p + m) .

Now it’s time for some mathematical teamwork. In Chapter 6, you proved The Bionomial Theorem.
Thanks to your work (good job, Reader! ) we know that

N N!
k
N
(p + m) = p m N−k .
k!(N − k)!
k=0
First of all, since p and m are natural numbers, and N and k are non-negative integers, each

N!
k
p m N−k > 0, foreach k =0, 1, 2,...,N.
k!(N − k)!

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