Page 208 - 'Blast_Into_Math

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

• Hint for # 7: Just have fun with this problem!
• Example for #8: First, let’s consider the case when x ≥ 0. In this case, all the terms in the
sequence
x n
x n = ≥ 0.
n!

So, we can prove that the series converges if we prove that the partial sums are all bounded above,
because then we can apply the Σ Theorem. Now, the series we know the best are the geometric
series. So, if we can compare this series to a geometric series, that would be helpful. Geometric
series are defined by a geometric sequence,

2
n
a 1 = a, a 2 = a ∗ a = a , andingeneral a n = a .
By the Geometric Σ Theorem, the geometric series converges precisely when |a| < 1. To
compare our series to a geometric series, remember that factorials are like onions, and when
we divide them by each other, we can remove layers of the onion. The ratio of the geometric
sequence is

a n+1
= a.
a n

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