Baccalauréat, série S

Session de juin 2007

Épreuve de section européenne Proof of the irrationality of

The number

e

can be dened has the innite sum

e=

e

∞ X 1 . n!

n=0 This denition can be used to prove that a fraction

e

is irrational, meaning that it can't be written as

a b.

We will prove this result by contradiction. Suppose that there exist two positive integers and

b

such that

e=

b X 1 x = b! e − n!

! = b!

n=0

We will rst show that

x

To see that

x

∞ X 1 . n!

n=b+1

is an integer, then show that

contradiction will establish the irrationality of

•

a

a b . Consider the number

x

is less than

1

and positive. The

e.

is an integer, note that

b X 1 x = b! e − n!

= b!

n=0 b X

a − b

n=0

= a(b − 1)! −

= a(b − 1)! −

= a(b − 1)! −

!

1 n!

!

b X b! n!

n=0 b X n=0 b X

1 · 2 · 3 · · · (n − 1)(n)(n + 1) · · · (b − 1)(b) 1 · 2 · 3 · · · (n − 1)(n) (n + 1)(n + 2) · · · (b − 1)(b).

n=0 Clearly, every term of this sum is an integer. Then, so is the number

•

From the second part of its denition it's clear that

0 < x.

x.

Moreover,

1 1 1 + + + ··· b + 1 (b + 1)(b + 2) (b + 1)(b + 2)(b + 3) 1 1 1 < + + + ··· b + 1 (b + 1)2 (b + 1)3 1 = b < 1.

x =

So

0 < x < 1.

Since there does not exist a positive integer less than so

e

1,

we have reached a contradiction, and

must be irrational. Adapted from Wikipedia.org, the free encyclopedia.

Questions 1. 2.

What other irrational numbers do you know ?

√

Suppose that the radical

√

such that

a. b. c. d. 3. 4. 5.

10 =

10

is rational. Then there exist two positive integers

m

and

n

m n.

What is the relation between

m2

and

n2 ?

What can you say about the number of zeros at the end of the square of any integer ? Find a contradiction.

√

What can you conclude about

10 ?

What is the meaning of the notation  n! ? Summarize in your own words the proof of the irrationality of

a.

Prove that for any positive integer

n X i=1

e.

n,

1 1 1 1 1 1 = + + ··· + = + i 2 3 n (b + 1) b + 1 (b + 1) (b + 1) (b + 1) b

b.

Deduce the limit of

c.

Where is this property used in the proof of the irrationality of

Pn

1 i=1 (b+1)i when

n

approaches



1 1− (b + 1)n

 .

+∞. e?

2007-08  Proof of the irrationality of e