MATH 1103 Homework 5Due Monday February 16, 2014

Problem 1. Suppose f(x) is any polynomial function.

a) Using integration-by-parts, show that∫ 1

0

f(x) sinπx dx =f(0)

π+f(1)

π− 1

π2

∫ 1

0

f ′′(x) sinπx dx.

b) Then replace f by f ′′ in part a) to show that∫ 1

0

f(x) sinπx dx =f(0)

π+f(1)

π− f ′′(0)

π3− f ′′(1)

π3+

1

π4

∫ 1

0

f (4)(x) sinπx dx.

c) Continuing like this, show that∫ 1

0

f(x) sinπx dx = F (0) + F (1),

where

F (x) =f(x)

π− f ′′(x)

π3+f (4)(x)

π5− · · ·

(which is a finite sum since f(x) is a polynomial).

Problem 2. Now suppose n is a positive integer and f(x) is a polynomial function of degree 2nall of whose derivatives f (k)(x) take integer values at x = 0 and x = 1. Prove that

π2n+1

∫ 1

0

f(x) sinπx dx

is an integer.

We are now ready to prove that π is irrational. In fact we will prove that π2 is irrational. (Thisimplies that π itself is irrational.) I will help you get started.

As in class, suppose π2 is rational. This would mean that

π2 =a

b(1)

for some integers a and b. Using the fact (proved in class) that

limn→∞

a2n

n!= 0,

1

we can choose n so thata2n

n!<

1

2.

From now on we work with such an n. We will apply exercise 1 and 2 to the polynomial function

f(x) =xn(1 − x)n

n!.

This function f(x) has the properties (proved in class) that

Property 1. 0 < f(x) < 1n!

for all x in the interval (0, 1).

Property 2. All derivatives of f take integer values at 0 and 1.

Now you take it from here.

Problem 3. a) Use Property 1 to show that

0 < πa2n∫ 1

0

f(x) sinπx dx < 1.

b) Use Property 2, along with (1) and the result of Problem 2, to show that

πa2n∫ 1

0

f(x) sinπx dx

is an integer.

c) Explain how the incompatibility of a) and b) implies that π is irrational.

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