Page 209 - 'Blast_Into_Math

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

Let’s look at the same ratio for the sequence

x n
x n = ,
(n +1)!

then

x n+1 n! x
x n+1
= = .
x n (n +1)! x n n +1

We know that x ∈ R , which means that x is either a rational number or the least upper
bound of a bounded set of rational numbers. In either case, there is some

m ∈ N such that m> x.

Then, if n ≥ 2m ,

x m 1
x n+1
= < < .
x n n +1 n +1 2

We can re-arrange this to

x n
x n+1 < , ∀n ≥ 2m.
2

Let’s think about this a bit more, and try to relate it further to a geometric series. We know that

x 2m x 2m+1 x 2m
x 2m+1 < , x 2m+2 < < .
2 2 2 2

Now you can prove by induction that

x 2m
x 2m+n < ∀n ∈ N.
2 n

So, now you can prove that the partial sums are all bounded, by splitting up the sum,

N n m n N n m n N−m
x x x x x m
S N = = + ≤ + , N ≥ m.
n! n! n! n! 2 n
n=0 n=0 m+1 n=0 n=1

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