Page 197 - 'Blast_Into_Math

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

(Hint: try a proof by induction.)

Therefore,

N N N
1 1 1

≤ ≤ ,
n! 2 n 2 n
n=2 n=2 n=1
and

N N N+1
1 1 1/2 − (1/2)
S N =1 +1 + ≤ 2+ =2 + < 3.
n! 2 n 1 − 1/2
n=2 n=1
Since x n > 0 for all n ∈ N , by the Σ Proposition, since the sequence of partial sums is bounded above
by 3, the series

∞
1

convergestothe least upper bound of thesequenceofpartial sums
n!
n=0
and we define this to be e,

∞
1
e := .
n!
n=0

Now that we have defined Euler’s constant e , we can define the natural logarithm, which plays a starring
role in the Prime Number Theorem.

Definition 7.5.1 For a real number x> 0, the natural logarithm of x , which we write as

log x,

is defined to be the unique real number y such that

y
e = x.

Remark 7.5.2 The natural logarithm is sometimes written as

ln x.

Why is there only one real number y such that

y
e = x?

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