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

is

x −x N+1 x x
+ lim = +0 = .
1 − x N→∞ 1 − x 1 − x 1 − x

x
x
Therefore, since the sequence of partial sums converges to 1−x , by definition, the series converges to 1−x .
What happens if |x|≥ 1? In Exercise # 4 at the end of this chapter you will prove that if a series,

∞

x n
n=1

converges then

lim x n =0.
n→∞

n ∞ converges to 0 if and only if
By the Geometric Sequence Lemma, the geometric sequence {x } n=1
|x| < 1. So if |x|≥ 1, then

n
lim x =0,
n→∞

Combining the geometric sequence Lemma together with your proof of Exercise # 4 at the end of this
chapter, we have proven the theorem using our mathematical teamwork.

♥
Exercise: Determine whether or not the following geometric series converge, and if they do, find their

limits.

1. ∞ 2 n .
n=1
2. ∞ (−2) n .
n=1
∞ −n .
3. n=1 3
4. ∞ 1 .
n=1 5 n

7.4 Decimal expansions

In Chapter 6, you learned how to write numbers in different bases. At the end of the chapter, you saw

how to write fractions in different bases. When we write a fraction in base 10, this is called the decimal
expansion of the fraction. In some cases, like , the decimal expansion goes on forever,
1
3
1
=0.33333333333.....
3

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