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

is a geometric sequence. When it converges, the series

∞

x n
n=1

is a geometric series. The number x is known as the ratio of the geometric series.

2
The first term in a geometric sequence is x . The next term is x = x ∗ x , which is x times the first
2
term. The next term is x = x ∗ x , so it is x times the previous term. Let’s visualize a geometric series
3
as a tower of similar looking mathematical ants. At the bottom, we have the first ant. Next, we have an
ant who looks exactly the same but is x times the size of the first ant. This ant comes along and stands
2
on top of the first one. Then, the third ant looks exactly like the other two, but it is x times as big. If
the ants keep getting bigger, this is going to be a problem for the first ant who is at the bottom of the

tower! But, if the ants keep getting smaller, then the first ant just might be able to hold up the whole
infinite tower of geometric ants.

To prove whether or not a geometric series converges we first need to prove an important fact about
geometric sequences.

Lemma 7.3.4 (Geometric Sequence Lemma). Let x ∈ R . Then the geometric sequence

∞ , x n = x n
{x n } n=1

converges to 0 if and only if |x| < 1.

Proof: The Lemma contains the phrase “if and only if” which means we must prove that two statements
are equivalent. The two statements are:

n
lim x =0,
n→∞

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