Page 185 - 'Blast_Into_Math

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

Let’s think about the meaning of a decimal expansion. Since we are in base 10, the digits are the integers
0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Each digit is the numerator of a fraction whose denominator is 10 raised
to some power. So, what 0.33333333... means is

3 3 3
+ + + ... .
10 10 2 10 3

This looks almost like a geometric series. Does the geometric series

1
∞ n

10
n=1

1
converge? By the Geometric Series Theorem, with 10 playing the role of x,

1 1
∞ n 1

= 10 = .
10 1 − 1 9
n=1 10
A series converges precisely when its sequence of partial sums converges. We can use the LAMP because

N N
3 1
T N = =3 =3S N ,
10 n 10 n
n=1 n=1

where S N are the partial sums of the geometric series,

N
1
S N = .
10 n
n=1
By the LAMP,

1 3 1
lim T N =3 lim S N =3 ∗ = = .
N→∞ N→∞ 9 9 3

Well, that’s not really surprising is it? Now, let’s prove that we can write any real number between 0 and
1 as a decimal expansion

∞
x n

, x n ∈{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
10 n
n=1

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