Page 210 - 'Blast_Into_Math

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

Then, for any N> m ,

N−m N N N+1
x m x m
1 1/2 − (1/2)
< = x m = x m <x m ,
2 n 2 n 2 n 1 − 1/2
n=1 n=1 n=1
m
x n

S N ≤ + x m , ∀N> m.
n!
n=0
Now, since the terms in the sequence

x n
x n = ≥ 0,
n!

by the Σ Theorem, the sequence of partial sums converges to its least upper bound.

We started by assuming that x ≥ 0. If x< 0, notice that

x n
> 0, for n even,
n!

x n
< 0, for n odd.
n!

You can split the partial sums into two parts, a positive part and a negative part,

N n
x
+
S N = = S + S ,
−
n! N N
n=0
+
where S is the sum of only the even terms, and S is the sum of only the odd terms. Note that
−
N N
N n N n
|x| |x|
+ −
S ≤ n! , S ≥− n! .
N
N
n=0 n=0
+
Since |x|≥ 0, we have already proven that S converges. Notice that
N
N n
|x|
−
−S ≤ n!
N
n=0

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