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Blast into Math! Analatic nummer theora: ants, ghosts and giants
7.5.1 Euler’s constant
Euler’s constant is one of the most important real numbers, which is known as e . This number was
discovered by Leonhard Euler. The number e is magical; it appears in every area of science. The number
e is also important in finance and economics. In number theory, we need e to understand how infinitely
many prime numbers there are.
If you write π in base 10, it has a decimal expansion which never ends. The same is true for e . There
are many equivalent ways to define e , but the simplest way for us to define e is the limit of a series.
First, we define the sequence
1
x n = ,
n!
with 0! defined to be equal to 1.
Exercise: Review your work on exercises # 9 in Chapter 4 and # 11 in Chapter 6. I hope this is not
becoming a mathematical under-the-bed monster.
Euler’s constant e is the limit of the series
∞
x n .
n=0
Why does this series converge? Because we can prove that it does!
Exercise: Prove that for all n ∈ N with n ≥ 2,
1 1
≤ .
n! 2 n
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