Page 173 - 'Blast_Into_Math

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

Proposition 7.3.2 (Σ Proposition). Let

∞
{x n } n=1

be a sequence of non-negative numbers. If the sequence of partial sums

N

∞ , S N =
{S N } N=1 x n
n=1

is bounded above, then the series converges to the least upper bound L of the sequence of partial sums,

∞

x n = L.
n=1

Proof: To prove this proposition, we will apply the Marching Ant Proposition to the sequence of partial
sums. The partial sums are monotonically increasing because

N N+1

S N = x n ≤ S N+1 = x n = S N + x N+1 , and x N+1 ≥ 0, ∀N ∈ N.
n=1 n=1

Since the partial sums are bounded above, they are a monotonically increasing sequence which is
bounded above. Therefore by the MAP, they converge to a limit, and that limit is the least upper bound
of the sequence of partial sums. By definition of convergence for series, the series also converges to the
same limit, and

∞

x n = L.
n=1

♥

7.3.1 Geometric series: a tower of similar-looking mathematical ants

In geometry, if a triangle has side lengths A , B , and C, and another triangle has side lengths 2A ,
2B, and 2C, what can we say about the two triangles? They are similar. The two triangles are the same

shape but have different sizes. You can take any shape and change its scale by multiplying each side
length by the same number.

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