1549901369-Elements_of_Real_Analysis__Denlinger_

7 .6 The Fundamental Theorem of Calculus 417

It is straightforward to prove (Exercise 19) that

F;:(x) = bnnZn+Z'l/Jn(x) -n^2 Fn(x)

= n^2 an'l/Jn(x) -n^2 Fn(x).

Thus, Fn satisfies the differential equation

Following the rules of elementary calculus, we have (Exercise 19),

(32)

(33)

(34)

d

dx [F~ (x) sin n x -n Fn(x) cos nx] =sin nx [F;: (x) + n^2 Fn(x)]. (35)

Combining Equations (34) and (35), we have

Thus,

fx [F~(x) sin nx - n Fn(x) cos nx] = n^2 an'l/Jn(x) sin nx.

1

1

n

2

an'l/Jn(x) sin nx dx = [ F~ (x) sin nx -n Fn(x) cos nx J:

= [F~ (1) sin n - n Fn(l) cos n] - [F~(O) sin 0 - n Fn(O) cos O]

= n [Fn(l) + Fn(O)].

Therefore, 1

1

nan'l/Jn(x) sin nxdx = Fn(l) + Fn(O), an integer.

However, by property (b) of 'l/Jn(x) given in Theorem 7.6. 20 ,

1

1

11 nan sin nx nan 11 nan

0 < nan'l/Jn(x) sin nxdx < 1 dx < - 1 dx = - 1 •

o o n. n. o n.

Now, lim na,n = 0 (see Corollary 2.3.11). Thus, by taking n sufficiently large,
n-+oo n.

0<1

1

nan'l/Jn(x) sin nxdx < 1.

As noted above, the integral appearing here is an integer. But it is impos-
sible to have an integer between 0 and 1. Therefore, n^2 cannot be rational; i.e.,
n^2 is irrational. Therefore, n is irrational. •

Concluding Remarks: The irrationality of n was established in 1767
by the Swiss mathematician Johann Heinrich Lambert, whose proof was made
more rigorous in 1794 by the French mathematician Adrien-Marie Legendre.
Their proofs used "continued fractions," which would take us too far afield to
discuss here.
The methods used here in proving Theorems 7 .6.21 and 7.6.22 are at-
tributed to the French mathematician Charles Hermite, 1873. These methods