Page 181 - 'Blast_Into_Math

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

and this we can re-arrange to

Nm 1
> − 1,
p

and finally get

p 1
N> − 1 .
m

So, we are free to choose the giant number N ∈ N to be any natural number

p 1
N> − 1 .
m

Then, since for all n ≥ N

N
p 1
n N
|x| ≤|x| ≤ < <,
q 1+ Nm/p
we have proven that for any > 0, there exists N ∈ N such that

n
|x| <, ∀n> N.
So, we have proven the second direction:

n
|x| < 1=⇒ lim x =0.
n→∞

♥

To prove whether or not a geometric series converges, we will use the Geometric Sequence Lemma and
a mathematical telescope.

Theorem 7.3.5 (Geometric Σ Theorem). Let x ∈ R, and define the geometric sequence

∞ n
{x n } n=1 , x n = x .

Then, if |x| < 1, the geometric series converges and

∞
x
n
x = .
1 − x
n=1
If |x|≥ 1, the geometric series does not converge.

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