Page 183 - 'Blast_Into_Math

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

does not change. What happens to the second part

−x N+1

1 − x

as N becomes large?

The denominator doesn’t change, only the numerator. By the Geometric Sequence Lemma since |x| < 1,
the sequence

∞ , x n = x n
{x n } n=1

−x
converges to 0. By the LAMP with 1−x playing the role of s ,

−x −x n+1 −x −x
n
n
t n = x = , lim t n = lim x = 0= 0.
1 − x 1 − x n→∞ 1 − x n→∞ 1 − x
x
Now we can use the LAMP again this time with 1−x playing the role of s , and the limit of the sequence

x −x N+1
∞
{S N } N=1 , S N = +
1 − x 1 − x

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